ACTION OF BRAID GROUPS ON DETERMINANTAL

IDEALS, COMPACT SPACES AND A

STRATIFICATION OF TEICHMÜLLER SPACE

Stephen P. Humphries

Abstract. We show that for n ≥ 3 there is an action of the braid group B_{n} on the determinantal ideals of a certain n × n symmetric matrix with algebraically independent entries off the diagonal and 2s on the diagonal. We show how this action gives rise to an action of B_{n} on certain compact subspaces of some Euclidean spaces of dimension \binom{n}{2}. These subspaces are real semi-algebraic varieties and include spheres of dimension \binom{n}{2}−1 on which the kernel of the action of B_{n} is the centre of B_{n}. We investigate the action of B_{n} on these subspaces. We also show how a finite number of disjoint copies of the Teichmüller space for the n-punctured disc is naturally a subset of this ℝ^{\binom{n}{2}} and how this cover (in the broad sense) of Teichmüller space is a union of non-trivial B_{n}-invariant subspaces. The action of B_{n} on this cover of Teichmüller space is via polynomial automorphisms. For the case n = 3 we show how to define modular forms on the 3-dimensional Teichmüller space relative to the action of B_{3}.

§1. Introduction.

The braid group B_{n} is the group with (standard) generators σ_{1}, …, σ_{n−1} and relations [Bi, p. 18]

σ_{i}σ_{i+1}σ_{i} = σ_{i+1}σ_{i}σ_{i+1}, for i = 1, …, n − 2,
σ_{i}σ_{j} = σ_{j}σ_{i},  for |i − j| > 1.

One of the main results of this paper is the following:

Theorem 1. For n ≥ 3 let U_{n} be the following symmetric matrix:

U_{n} = ( \begin{matrix}2 & a_{21} & … & a_{i1} & … & a_{n−1 1} & a_{n1} \\ a_{21} & 2 & … & a_{i2} & … & a_{n−1 2} & a_{n2} \\ ⋮ & ⋮ & ... & ⋮ & & ⋮ & ⋮ \\ a_{i1} & a_{i2} & … & 2 & … & a_{n−1 i} & a_{ni} \\ ⋮ & ⋮ & & ⋮ & ... & ⋮ & ⋮ \\ a_{n−1,1} & a_{n−1,2} & … & a_{n−1 i} & … & 2 & a_{n n−1} \\ a_{n1} & a_{n2} & … & a_{n i} & … & a_{n,n−1} & 2 \end{matrix} )

where the a_{ij}, 1 ≤ j < i ≤ n, are algebraically independent indeterminates generating a polynomial ring R′_{n} over a commutative ring R with identity. The ring R′_{n} is acted upon by B_{n} with kernel the cyclic centre Z(B_{n}) of B_{n}.

Typeset by AMS-TEX

For r ≥ 0 let I_{nr} be the determinantal ideal (of R′_{n}) generated by all of the (r+1)×(r+1)- minors of U_{n}. Then the braid group acts on each I_{nr}.

The braid group also acts on the quotients R′_{n}/I_{nr}.

(i) Suppose that 2 is invertible in R. Then for n > 3 the kernel of the action of B_{n}/Z(B_{n}) on R′_{n}/I_{n 1} contains the non-trivial normal subgroup generated by (σ_{1})^{4}. The action of B_{n}/Z(B_{n}) on R′_{n}/I_{n1} determines an epimorphism B_{n} → W (D_{n}) where W (D_{n}) is the Coxeter group of type D_{n}. (ii) Suppose that 2 is invertible in R. Then for n > 3 the kernel of the action of B_{n}/Z(B_{n}) on R′_{n}/I_{n 2} contains the non-trivial normal subgroup generated by (σ_{1}σ_{2})^{6}. (iii) For any commutative ring R the action of B_{n}/Z(B_{n}) on R′_{n}/I_{n n−1} has trivial kernel. (iv) For R = ℤ or R a field having characteristic 2, the action of B_{n}/Z(B_{n}) on R′_{n}/I_{n r} has trivial kernel for all 1 ≤ r < n.

We will also show how this result enables us to find various compact subsets of ℝ^{\binom{n}{2}} which are acted upon by B_{n}. We then show that certain subsets of this ℝ^{\binom{n}{2}} can be identified with a finite number of disjoint copies of the Teichmüller space for the punctured disc and combining this with results of [H2] we will obtain a non-trivial stratification of this cover (in the broad sense) of Teichmüller space by B_{n}-invariant subsets. We now explain where this action of B_{n} on ℝ^{\binom{n}{2}} comes from.

Let D_{n} be the disc with n punctures π_{1}, …, π_{n}. Then B_{n} acts as (isotopy classes of) diffeomorphisms of D_{n} [Bi, Ch.1]. Further, for 2 ≤ m ≤ n, B_{n} acts transitively on the set of isotopy classes of positively oriented simple closed curves on D_{n} which surround m of the punctures. The generator σ_{i} acts as a half-twist [Bi] on D_{n} interchanging π_{i} and π_{i+1} and has a representative diffeomorphism which is supported in a tubular neighbourhood of an arc a_{i} joining π_{i} to π_{i+1} (see Figure 1).

In Figure 1 we have shown the arcs a_{i}. This fact allows one to construct [Bi] a faithful representation of B_{n} as automorphisms of a free group F_{n} =< x_{1}, …, x_{n} >, which we identify with the fundamental group π_{1}(D_{n}). Here the x_{i} are (homotopy classes of) simple closed curves surrounding one of the punctures and based at a fixed point of the boundary of D_{n}, as in Figure 1. A characterisation of the image of B_{n} in Aut(F_{n}) was given by Artin as follows: ϕ ∈ Aut(F_{n}) is the image of a braid if and only if (i) for all 1 ≤ i ≤ n, ϕ(x_{i}) is a conjugate of some x_{j}; and (ii) ϕ(x_{1}x_{2} … x_{n}) = x_{1}x_{2} … x_{n}. The action of B_{n} on the generators x_{i} is as follows: let 1 ≤ j < n; then

σ_{j}(x_{i}) = x_{i} if i ≠ j, j + 1, σ_{j}(x_{j}) = x_{j}x_{j+1}x_{j}^{−1}, σ_{j}(x_{j+1}) = x_{j}.

Let R be a commutative ring with identity and let

R_{n} = R[a_{12}, a_{13}, …, a_{1n}, a_{21}, a_{23}, …, a_{2n}, …, a_{n1}, …, a_{n n−1}]

be a polynomial ring in commuting indeterminates a_{ij}, 1 ≤ i ≠ j ≤ n. It will be convenient to put a_{ii} = 0 for all i ≤ n. In a previous paper [H1] we have shown that by representing the free group π_{1}(D_{n}) using transvections (see below) and looking at certain traces we obtain an action of B_{n} on the ring R_{n} i.e. we have a homomorphism

ψ_{n} : B_{n} → Aut(R_{n});

the kernel of ψ_{n} is the centre of B_{n} [H1]. This is understood as follows.

For fixed n ≥ 2 we let Π_{n} = T_{1}T_{2} … T_{n} where the T_{i} are certain n×n matrices (transvections):

T_{i} = ( \begin{matrix}1 & 0 & … & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & … & ⋮ & ⋮ \\ a_{i1} & a_{i2} & … & 1 & … & a_{in−1} & a_{in} \\ ⋮ & ⋮ & … & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 0 & … & 1 & 0 \\ 0 & 0 & … & 0 & … & 0 & 1 \end{matrix} ) .

Here the non-zero off-diagonal entries of T_{i} occur in the ith row. One way of defining a transvection [A] is as a matrix M = I_{n} + A where I_{n} is the n × n identity matrix, det(M) = 1, rank(A) = 1 and A^{2} = 0. In particular, conjugates of transvections are transvections. That < T_{1}, …, T_{n} > is a free group of rank n is shown in [H2]. This allows us to identify x_{i} and T_{i} for i = 1, …, n and so to identify F_{n} and < T_{1}, …, T_{n} > .

Now, since we are identifying x_{i} ∈ F_{n} with T_{i} ∈< T_{1}, …, T_{n} >, we see, by Artin’s characterisation (i) and (ii) above, that the matrix Π_{n} is invariant under the action of B_{n} and so that Π_{n} + λI_{n} is also invariant under the action of B_{n} for any choice of λ ∈ R. Thus the action fixes the various determinantal ideals I_{n,r,λ} determined by Π_{n} + λI_{n}. Here I_{n,r,λ} is the ideal generated by the determinants of all (r + 1) × (r + 1) submatrices of Π_{n} + λI_{n}. These are the points at which Π_{n} + λI_{n} has rank r (except in the characteristic 2 case). For general properties of such rings see [BV, DEP1, DEP2]. We will put I_{n,r} = I_{n,r,1}.

Now, since Π_{n} is invariant under the action of B_{n}, it follows that the characteristic polynomial

χ_{n}(x) = ∑_{i=0}^{n} c_{ni}x^{i}

of the matrix Π_{n} has coefficients c_{ni} = c_{ni}(a_{12}, …, a_{n n−1}) ∈ R_{n} which are invariant under the action of B_{n} (this was first noted in [H2, Theorem 2.8]). We there also noted that the c_{ni} are non-homogeneous polynomials of degree n for 1 ≤ i < n. If n < 6, then they generate the ring of invariants for the action of B_{n} on the subring Y_{n} of R_{n} defined in the next paragraph [H3]. Note that we have c_{nn} = 1 and c_{n0} = ±1.

For i, j, k, …, r, s ∈ {1, 2, …, n} let c_{ijk...rs} denote the cycle a_{ij}a_{jk} … a_{rs}a_{si} ∈ R_{n}. Then the cycles generate a subalgebra of R_{n} denoted Y_{n}. A cycle c_{ijk...rs} will be called simple if i, j, k, ..., r, s are all distinct. The ring Y_{n} is generated by the (finite number of) simple cycles and is B_{n}-invariant.

The representation of B_{n} in Aut(R_{n}) can be thought of in the following way. Note that the action of B_{n} on D_{n} fixes the boundary of D_{n}. This fact gives Artin’s condition (ii) above. Let C_{n} denote the set of all oriented simple closed curves on D_{n}. Now, from the above, B_{n} acts by automorphisms on F_{n} in such a way that for α ∈ B_{n} the matrix α(T_{i}) is a conjugate of some T_{j}, 1 ≤ j ≤ n i.e. α(T_{i}) is also a transvection. Further, if c ∈ C_{n}, then c represents a conjugacy class in F_{n} and so its trace is well-defined (the trace of the corresponding product of transvections in F_{n} =< T_{1}, …, T_{n} >). In fact one easily sees that trace(c) ∈ Y_{n} [H1]. Then a map ϕ = ϕ_{n} : C_{n} → R_{n} is defined by

ϕ_{n}(c) = trace(c) − n.

Thus ϕ_{n} can be thought of as being defined on certain conjugacy classes of elements of F_{n} (namely those representing simple closed curves). The map ϕ can be extended to act on all of F_{n}, by the requirement that for s ∈ F_{n} we have ϕ(s) = trace(s) − n. It is easy to see that if w = T_{i_{1}}^{e_{1}} … T_{i_{r}}^{e_{r}} ∈< T_{1}, …, T_{n} > is cyclically reduced as written with e_{i} ≠ 0, i_{k} ≠ i_{k+1}, i_{r} ≠ i_{1}, and r > 1, then ϕ(w) is a polynomial in Y_{n} of degree r (see §2).

Now for m ≥ n and s ∈ F_{n} we may also consider s as an element of F_{m} under the natural inclusion of F_{n} in F_{m}. In this case we note that ϕ(s) has the same value whether we consider s as an element of F_{n} or F_{m}.

A fundamental property of the transvections T_{i} is that for all 1 ≤ i, j ≤ n we have trace(T_{i}T_{j}) = a_{ij}a_{ji} + n and in general if A, B ∈ F_{n}, then

trace(AT_{i}A^{−1}BT_{j}B^{−1}) = b_{ij}b_{ji} + n,  (1.1)

where b_{ij} ∈ R_{n} (see [H1]). It is also easy to see that there is a natural choice so that

b_{ij} = ±a_{ij} + terms of higher degree.

For example trace((T_{1}T_{2}T_{1}^{−1})T_{3}) = (a_{23} − a_{21}a_{13})(a_{32} + a_{31}a_{12}) + n. This is explained in detail in §2.

Now for α ∈ B_{n} the image α(T_{i}) is a conjugate AT_{j}A^{−1} for some A ∈ F_{n} and 1 ≤ j ≤ n (by Artin’s condition (i) above). Here the action of α on the a_{ij} is defined by

ϕ(α(T_{i})α(T_{j})) = α(a_{ij})α(a_{ji}),

(see §2 for more details) so that it has the following naturality property (with respect to the action of B_{n} on F_{n}): for all w ∈< T_{1}, …, T_{n} >, α ∈ B_{n}, we have

ϕ(α(w)) = α(ϕ(w)).

For example the action of the generator σ_{i}, 1 ≤ i < n, is given by

σ_{i}(a_{i i+1}) = a_{i+1 i}, σ_{i}(a_{i+1 i}) = a_{i i+1},
σ_{i}(a_{h i}) = a_{h i+1} + a_{hi}a_{i i+1}, σ_{i}(a_{h i+1}) = a_{h i},  (1.2)
σ_{i}(a_{i h}) = a_{i+1 h} − a_{i+1 i}a_{ih}, σ_{i}(a_{i+1 h}) = a_{i h},

where 1 ≤ h ≤ n and h ≠ i, i + 1.

It follows from [H1, Theorem 2.5 and Theorem 6.2] that the kernel of the action of B_{n} on R_{n} is the centre of B_{n} and that if B_{n} and R_{n} are thought of as sub-objects of B_{n+1} and R_{n+1} (respectively), then the action of B_{n} on R_{n+1} is faithful.

We note as in [H2] that there is a natural ring involution ∗ on R_{n} which commutes with the action of B_{n}, so that for α ∈ B_{n} we have

α(w)^{∗} = α(w^{∗})  (1.3)

for all w ∈ R_{n}. This involution is determined by its action on the generators a_{ij} which is as follows:

a_{ij}^{∗} = −a_{ji}.

(Thus to check (1.3) one need only consider the situation where α = σ_{i} and w = a_{rs}.) This involution has the following property:

trace(A^{−1}) = trace(A)^{∗},

for all A ∈ F_{n}. Thus for c ∈ C_{n} we have ϕ(c^{−1}) = ϕ(c)^{∗}, where c^{−1} is the curve c with its orientation reversed. We also have b_{ji} = −b_{ij}^{∗}, for b_{ij}, b_{ji} as in (1.1).

Now factoring out by the action of the above involution leads us to consider the situation where a_{ij} = −a_{ji} for all 1 ≤ i ≠ j ≤ n. Let R′_{n} denote the corresponding quotient of R_{n} and T′_{i} the corresponding transvections etc. The ring R′_{n} is isomorphic to the subring of R_{n} generated by the a_{ij} with i > j and so is a polynomial ring. Then B_{n} also acts on R′_{n}. The fact that < T′_{1}, …, T′_{n} > is still a free group and that the kernel of the action of B_{n} on R′_{n} is still the centre Z(B_{n}) were noted in [H1, H2]. We will let Π′_{n} = T′_{1}T′_{2} … T′_{n} and

χ′_{n}(x) = ∑_{i=0}^{n} c′_{ni}x^{i}

the characteristic polynomial of the matrix Π′_{n}. Again the coefficients c′_{ni} are invariant under the action of B_{n} on R′_{n}.

We will primarily be interested in the situation where R = ℝ (due in part to the connection with Teichmüller space); however the presence of the 2s on the diagonal of U_{n} will force us to distinguish between rings R where 2 is invertible or where 2 is not invertible. In the case R = ℝ each ideal I_{n,r,λ} determines a real algebraic variety V_{n,r,λ} ⊂ ℝ^{\binom{n}{2}}. These varieties are invariant under the action of B_{n}. For general properties of real algebraic varieties see [BCR]. Another reason for looking at the case R = ℝ is that in this case there is a compact piece of V_{n,r} = V_{n,r,1} which is also B_{n}-invariant. We will determine the nature of this compact set. Recall that a semi-algebraic set in ℝ^{n} is (roughly speaking) a set of points determined by a set of algebraic equalities and inequalities (including all finite unions of finite intersections of such). For example we prove Theorem 2. Suppose that R = ℝ. For all k ≤ n there is a compact real algebraic set V_{n,k,1}^{(2)} ⊂ V_{n,k,1} which is invariant under the above action of B_{n}.

For n = 3 the semi-algebraic subset V_{3,2,1}^{(2)} is homeomorphic to a 2-sphere, and is smooth except at 4 singular points.

For n ≥ 4 the semi-algebraic subset V_{n,2,1}^{(2)} is homeomorphic to the quotient

S^{1} × S^{1} × ⋯ × S^{1}/α,

where α is the inverse map

α(z_{1}, …, z_{n−1}) = (z_{1}^{−1}, …, z_{n−1}^{−1}),

on (S^{1})^{n−1}. Here V_{n,2,1}^{(2)} has 2^{n−1} singular points and is otherwise smooth. For n = 4 we find a presentation for the image of B_{4} in Aut(R′_{4}/I_{4,2,1}) and show that this group has a faithful 3 × 3 linear representation over ℂ.

For all n ≥ 3 the group B_{n} acts on a one-parameter family of smooth topological spheres of dimension \binom{n}{2} − 1, each such sphere being B_{n}-invariant. These spheres come from level sets of det(U_{n}). Each point of these spheres corresponds naturally to a positive definite symmetric matrix. For a dense set of the parameter values the kernel of this action of B_{n} on the corresponding sphere is the cyclic centre Z(B_{n}).

When n = 4 there are two convex 5-balls in ℝ^{6} the boundary of whose intersection is a B_{4}-invariant 4-sphere.

More details of these varieties and the B_{n}-actions will be given in the rest of this paper. In the following we will need to recall [Bi] that there is an epimorphism π : B_{n} → S_{n},

where S_{n} is the symmetric group on n objects, which is induced by sending σ_{i} to the transposition (i, i + 1) ∈ S_{n} for each i = 1, …, n − 1. The kernel of π is P_{n} the group of pure or coloured braids on n strings.

In relation to the action of B_{n} on the determinantal rings indicated in Theorem 1 we should also mention the paper [H4] where we show that B_{4} acts on an ordinal Hodge algebra or algebra with straightening law. One hope is that using either approach we may be able to say something about the representation theory of B_{n} or P_{n} using (standard) methods as in [BV, DEP1, DEP2, JPW]. The present paper is more topological in nature, however we investigate the representation theory of the action given above more fully in [H5].

Suppose that R contains the rational numbers. Now in [H3] we proved that for any α ∈ P_{n} there is a derivation D(α) of the power series algebra R̄_{n} of R_{n} such that

α(x) = exp(D(α))(x)

for all x ∈ R̄_{n}. For example we showed that

D(σ_{1}^{2}) = \begin{matrix}arcsinh & (\sqrt{a_{12}a_{21} + (a_{12}a_{21})^{2}/4}) \\ √ & \overline{a_{12}a_{21} + (a_{12}a_{21})^{2}/4} \end{matrix}((a_{32}a_{21} + a_{31}a_{12}a_{21}/2)\frac{∂}{∂a_{31}}

+ (−a_{12}a_{23} + a_{12}a_{21}a_{13}/2)\frac{∂}{∂a_{13}} + (a_{31}a_{12} − a_{32}a_{21}a_{12}/2)\frac{∂}{∂a_{32}}

+ (−a_{21}a_{13} − a_{21}a_{12}a_{23}/2)\frac{∂}{∂a_{23}} ).

These derivations D(α) have the property that D(α)(c_{ni}) = 0 for all i ≤ n (this being equivalent to the invariance of the c_{ni}). We obtain a group

P_{n} =< exp(tD(α))|α ∈ P_{n}, t ∈ ℝ > ⊂ Aut(R̄_{n})

using composition of functions, or equivalently, using the Campbel-Baker-Hausdorff formula [J]. We can extend P_{n} to

B_{n} =< P_{n}, σ_{1}, …, σ_{n−1} > ⊂ Aut(R̄_{n})

to obtain a continuous group with B_{n}/Z(B_{n}) as a subgroup. We will prove:

Theorem 3. Suppose that R contains the rational numbers. Then the group B_{n} acts on each of the spaces V_{nk} and V_{nk}^{(2)} (if R = ℝ) and also on the level sets of the functions c_{ni}.

We assume that R = ℝ for the following discussion, so that topologically we have R′_{n} ≅ ℝ^{\binom{n}{2}}. The action of B_{n} on R′_{n} comes, as described above, from considerations of trace algebras i.e. character varieties. The motivation for looking at actions of groups on character varieties was to give coordinates for Teichmüller space. This approach originated with Fricke and Klein [FK] and there have been many subsequent attempts at ways of giving these real analytic trace coordinates (see for example [K1, K2, O, Sa] and references therein). Thus it is not surprising to see that there is a connection with our invariant varieties and Teichmüller space, which we now describe.

Let G < P SL_{2}(ℝ) be a Fuchsian group acting discontinuously on the upper half plane ℍ and such that ℍ/G is conformally equivalent to a Riemann surface of genus g with n ordered points and m conformal discs removed; we say that G has type (g, n, m). We choose a marking on G by specifying a set of generators a_{1}, b_{1}, …, a_{g}, b_{g}, c_{1}, …, c_{n}, e_{1}, …, e_{m} for G. For example one can do so such that the only relator is

e_{m} … e_{1}c_{n} … c_{1}b_{g}^{−1}a_{g}^{−1}b_{g}a_{g} … b_{1}^{−1}a_{1}^{−1}b_{1}a_{1} = id.

There is a non-degeneracy condition that is required for the following to work, namely that 6g + 3m + 2n − 6 > 0. Any two marked Fuchsian groups are conformally equivalent if they are conjugate in P SL_{2}(ℝ). The set of such equivalence classes of Fuchsian groups of type (g, n) is called the Teichmüller space of type (g, n, m) and is denoted by T_{g,n,m}. It is well known that T_{g,n,m} is a real analytic manifold of dimension 6g + 3m + 2n − 6 [Ab]. Taking traces of a finite number of products of the images of these generators results in a (finite) set of real analytic coordinates for T_{g,n,m} [FK, K1, K2, O]. In our case we are interested in the situation where g = 0, m = 1. Usually one associates an element ν_{i}, i = 1, …, n, of the set {2, 3, …, ∞} to each deleted point, however we will only be interested in the situation where ν_{i} = ∞ for all i = 1, …, n. We also note that each c_{i} is a parabolic element (its squared trace is 4) and that any non-identity element not conjugate to some power of some c_{i} is hyperbolic (has squared trace greater than 4). Now from the discussion of transvections above it appears natural to solve

trace(c_{i}c_{j}) = 2 − a_{ij}^{2}  (1.4)

for all 1 ≤ j < i ≤ n. We will show that the B_{n} actions on both sides of the equation \sqrt{2 − trace(c_{i}c_{j})} = ±a_{ij} are compatible at least for some choices of the sign of ±a_{ij}. We thus obtain a representation of Teichmüller space as a subspace of our coordinate space ℝ^{\binom{n}{2}}. Actually we get a finite disjoint cover of Teichmüller space corresponding to some of the choices of sign that we get when solving (1.4), so that each connected component of this cover is isomorphic to Teichmüller space via the covering projection.

For n = 3, 4 it will follow from dimensional considerations that T̄_{0,n,1} has a non-trivial stratification by B_{n}-invariant subsets. For n ≥ 5 we see that T̄_{0,n,1} has dimension 2n − 3; however there are at most ⌊\frac{n}{2}⌋ independent invariants: c′_{51}, …, c′_{5⌊ \frac{n}{2} ⌋} (see Lemma 2.5). Since

T̄_{0,n,1} ⊂ ℝ^{\binom{n}{2}} it does not follow from such dimensional considerations that we have a nontrivial stratification (by level sets of the c_{ni}) in this case. However we will show that there actually is such a non-trivial stratification by showing that c′_{n n−1} is not constant on each component of T̄_{0,n,1}. We do this by exhibiting a 1-parameter family of matrices generating a 1-parameter family of Fuchsian groups, each giving a point of Teichmüller space and showing that c′_{n n−1} is not constant on any of the lifts of this family to T̄_{0,n,1}.

Theorem 4. There is a real analytic subset T̄_{0,n,1} of ℝ^{\binom{n}{2}} which is a union of 2^{n−1} disjoint copies of the Teichmüller space T_{0,n,1} of the punctured disc D_{n}. The set T̄_{0,n,1} is a union of B_{n}-invariant pieces (strata) corresponding to the level sets of the invariants c′_{ni}. The strata corresponding to the invariant c′_{n1} = ±c′_{n n−1} are of codimension 1 in T̄_{0,n,1}.

For n = 3 we determine a fundamental domain for the action of B_{3}/Z(B_{3}) on certain of the 2-dimensional level sets in ℝ^{3} of the invariant function c′_{31}. For t < 4 and |a_{ij}| > 2 there are 4 connected components of the level set c′_{31} = t and we may identity them using the identifications (x, y, z) ≡ (−x, −y, z) ≡ (−x, y, −z) (see §11). With this understood we have the following result which says that the vanishing of the gradient of the invariant function c′_{31} cuts out a region which is a fundamental domain for the action of a finite index subgroup of B_{3}/Z(B_{3}) ≅ P SL(2, ℤ) on the level surfaces of c′_{31}, thus indicating a close relationship between the group and the level surfaces:

Theorem 5. Consider the action of B_{3}/Z(B_{3}) on ℝ^{3} = ℝ[a_{21}, a_{31}, a_{32}]. For t < 0 there is a fundamental domain for the action of a subgroup H_{3} < B_{3}/Z(B_{3}) of index 3 on the level sets

c′_{31} = a_{21}^{2} + a_{31}^{2} + a_{32}^{2} − a_{21}a_{31}a_{32} = t.

In fact for such values of t the functions

\frac{∂c′_{31}}{∂a_{21}} = 2a_{21} − a_{31}a_{32}, \frac{∂c′_{31}}{∂a_{31}} = 2a_{31} − a_{21}a_{32}, \frac{∂c′_{31}}{∂a_{32}} = 2a_{32} − a_{31}a_{21},

cut out a region of the level set c′_{31} = t which is a fundamental domain for this action.

The subgroup H_{3} is freely generated by 3 involutions. Each of the curves determined by the above equations is fixed by one of these three involutions.

See Figure 8 for a representation of this fundamental domain. In proving the above result we give a natural generalisation of results relating to the Markoff equation and the Markoff tree [CF, Mo].

As indicated above there are various kinds of “natural” coordinates that can be defined on Teichmüller space, including such coordinates coming from traces of various products of generators. Relative to these coordinates the action of the mapping class groups (in our case the braid groups) is at best via rational maps (in equation (11.4) we have written down such rational maps for the action of the generators of B_{4} for some such choice of coordinates). However we now have the following indication that the a_{ij} also give a very natural set of coordinates for T_{0,n,1}:

Theorem 6. There is an embedding of a disjoint union of 2^{n−1} copies of the Teichmüller space in a Euclidean space ℝ^{\binom{n}{2}} (as in Theorem 4) and the action of B_{n} on this cover is via polynomial automorphisms.

In §2 we give more details on the action of B_{n} on R_{n} and R′_{n}. In §3 we consider the rank 1 case. In §4 we investigate the case where n = 3. In §5 we give various results on symmetric matrices that will be used later. In §6 we investigate the case where n = 4. In §7 we look at the rank 2 case. §8 is devoted to the rank n − 1 case. In §9 we study faithfulness questions for the action of B_{n} on various quotients of R′_{n}. In §10 we prove Theorem 3. The part of this paper concerning Teichmüller space is to be found in §11 and is largely independent of §§5 − 10.

§2 Action of B_{n} on R_{n} continued

In this section we describe in greater detail the action of B_{n} on R_{n} and on R′_{n} so as to be able to give explicit formulae for the action of certain braids. In general [A] a transvection in SL(Q^{n}) (for a commutative ring Q with identity) can be defined as a pair T = (ϕ, d) where d ∈ Q^{n} and ϕ is an element of the dual space of Q^{n} satisfying ϕ(d) = 0. The action is given by

T (x) = x + ϕ(x)d for all x ∈ Q^{n}.

Then we have [H1, Lemma 2.1]

Lemma 2.1. Let T = (ϕ, d) and U = (ψ, e) be two transvections. Then for all λ ∈ ℤ we have

U^{λ}T U^{−λ} = (ϕ − λϕ(e)ψ, U^{λ}(d)). □

Let T = {T_{1} = (ϕ_{1}, d_{1}), …, T_{n} = (ϕ_{n}, d_{n})} be a fixed set of transvections in SL((R_{n})^{n}) where ϕ_{i}(d_{j}) = a_{ij} for all 1 ≤ i ≠ j ≤ n as in the above. For any set of transvections

T′ = {T′_{1} = (ϕ′_{1}, e′_{1}), …, T′_{n} = (ϕ′_{n}, e′_{n})}

we let M(T′) denote the n × n matrix (ϕ′_{i}(e′_{j})) and we call M(T′) the M-matrix of the set of transvections T′.

Any monomial in R_{n} that can be written in the form a_{j_{1}j_{2}}a_{j_{2}j_{3}} … a_{j_{r−2}j_{r−1}}a_{j_{r−1}j_{r}} (with j_{i} ≠ j_{i+1} for 1 ≤ i < r) will be called a j_{1}j_{r}-word. Note that by (1.2) if α ∈ B_{n} and 1 ≤ i ≠ j ≤ n, then α(a_{ij}) is a sum of rs-words, where α(T_{i}) is a conjugate of T_{r} and α(T_{j}) is a conjugate of T_{s}. Let α ∈ B_{n} where α(T_{i}) = w_{i}T_{j}w_{i}^{−1} in freely reduced form for i = 1, ..., n and where w_{i} = w_{i}(T_{1}, …, T_{n}). Then for i = 1, …, n we have w_{i}T_{i}w_{i}^{−1} = (ψ_{i}, f_{i}) for some ψ_{i}, f_{i} determined by Lemma 2.1, which result in fact shows that

ψ_{i} = q_{1}ϕ_{1} + ⋯ + q_{n}ϕ_{n} and f_{i} = p_{1}d_{1} + ⋯ + p_{n}d_{n},  (2.1)

where p_{1}, …, p_{n}, q_{1}, …, q_{n} ∈ R_{n}. Since the a_{ij} are algebraically independent the ϕ_{i} and d_{j} are linearly independent and so the above representation is unique. We define the action of B_{n} on R_{n} by

α(a_{ij}) = ψ_{i}(f_{j}).

One can check that this agrees with the previous definition. Thus the M-matrix is acted upon naturally by B_{n}:

α(M(T)) = M({α(T_{1}), …, α(T_{n})}).

From Lemma 2.3 of [H1] we have:

Lemma 2.2. Let α ∈ B_{n} where α(T_{i}) = C_{1}T_{k}C_{1}^{−1}, α(T_{j}) = C_{2}T_{p}C_{2}^{−1}, with C_{1}, C_{2} ∈ < T_{1}, …, T_{n} > and let C = C_{1}^{−1}C_{2} = T_{j_{1}}^{q_{1}} … T_{j_{r}}^{q_{r}} be freely reduced with j_{r} ≠ p, j_{1} ≠ k, q_{s} ≠ 0 for s = 1, …, r and j_{s} ≠ j_{s+1}, for s = 1, …, r − 1. Then

α(a_{ij}) = ∑_{h=1}^{n} A_{h}a_{hp}

where A_{h} is equal to the sum of all the products of the form

q_{r_{1}}q_{r_{2}} … q_{r_{m}}a_{kj_{r_{1}}} a_{j_{r_{1}} j_{r_{2}}} … a_{j_{r_{m−1}} j_{rm}}

where 1 ≤ r_{1} < r_{2} < ⋯ < r_{m} ≤ r and j_{r_{m}} = h. If p ≠ j_{r}, then the summand of α(a_{ij}) of highest degree is unique and is equal to

±q_{1}q_{2} … q_{r}a_{kj_{1}}a_{j_{1}j_{2}} … a_{j_{r−1}j_{r}}a_{j_{r}p}. □

For example, if α(T_{1}) = T_{3}T_{2}^{−1}T_{1}T_{2}T_{3}^{−1} and α(T_{2}) = T_{2}^{−1}T_{3}T_{2}, then we would have C = T_{2}T_{3}^{−1}T_{2}^{−1} and

α(a_{12}) = a_{13} + a_{13}a_{32}a_{23} + a_{12}a_{23}a_{32}a_{23}.

We showed in [H1, Lemma 2.10] that for α ∈ B_{n} the freely reduced form of α(T_{i}) ∈< T_{1}, …, T_{n} > has no subword of the form T_{j}^{±2}. In fact if c ∈ C_{n} is a simple closed curve, then any cyclically reduced word in < T_{1}, …, T_{n} > which represents c has no subword of the form T_{j}^{±2}. This allows one to sharpen the conclusion of Lemma 2.2:

Lemma 2.3. If α ∈ B_{n} and 1 ≤ i ≠ j ≤ n, then the coefficient of the unique monomial of highest degree in α(a_{ij}) is ±1. □

One can be more specific about the coefficients in (1.2), namely

Lemma 2.4. Let C ∈< T_{1}, …, T_{n} >. Then CT_{i}C^{−1} = (ψ, f), where ψ = ∑_{i} λ_{i}ϕ_{i}, f = ∑_{i} μ_{i}d_{i}, where λ_{i}, μ_{i} ∈ R_{n} satisfy λ_{i}^{∗} = μ_{i} for all i.

Proof. This uses Lemma 2.1 and is by induction on the length of C in the standard generators T_{1}, …, T_{n}. □

Thus if i ≠ j and A = C_{1}T_{i}C_{1}^{−1} = (ψ_{1}, f_{1}), B = C_{2}T_{j}c_{2}^{−1} = (ψ_{2}, f_{2}) ∈< T_{1}, …, T_{n} >, then f_{1}, f_{2} are linearly independent and relative to this basis we have

A = ( \begin{matrix}1 & ψ_{1}(f_{2}) \\ 0 & 1 \end{matrix} ) , B = ( \begin{matrix}1 & 0 \\ ψ_{2}(f_{1}) & 1 \end{matrix} ) ,

where by Lemma 2.4 we have

ψ_{1} = ∑_{i} λ_{i}ϕ_{i}, f_{1} = ∑_{i} λ_{i}^{∗}d_{i}, ψ_{2} = ∑_{i} μ_{i}ϕ_{i}, f_{2} = ∑_{i} μ_{i}^{∗}d_{i}.

Thus trace(AB) = 2 + ψ_{1}(f_{2})ψ_{2}(f_{1}) = 2 + ϕ(AB) and we have

ψ_{1}(f_{2}) = ∑_{i,j=1}^{n} λ_{i}μ_{j}^{∗}ϕ_{i}(d_{j}), ψ_{2}(f_{1}) = ∑_{i,j=1}^{n} μ_{j}λ_{i}^{∗}ϕ_{j}(d_{i}).

Now let us refer back to (1.1). Since ϕ_{i}(d_{j}) = a_{ij} we see that if we let b_{ij} = ψ_{1}(f_{2}), b_{ji} = ψ_{2}(f_{1}), that we have trace(AB) = 2 + b_{ij}b_{ji} and

b_{ij}^{∗} = ψ_{1}(f_{2})^{∗} = −ψ_{2}(d_{1}) = −b_{ji},

thus proving (1.1) and the relation b_{ij}^{∗} = −b_{ji}.

Lemma 2.5. For all i ≤ n we have c′_{ni} = ±c′_{n n−i}.

Proof. By [Hu2, Corollary 2.7] we see that the characteristic polynomial χ′_{n}(x) satisfies

χ′_{n}(x) = (−x)^{n}χ′_{n}(1/x)^{∗},

so that, up to a sign, the list c′_{n1}, c′_{n2}, …, c′_{n n−2}, c′_{n n−1} is the same forwards and backwards when we are in the ring R′_{n} where a_{ij} = −a_{ji}. □

§3 The rank 1 case

We first note that the matrix U_{n} can never have rank 0.

Lemma 3.1. The i, j entry of Π_{n} is the sum of all monomials of the form

a_{i_{1}i_{2}}a_{i_{2}i_{3}}a_{i_{3}i_{4}} … a_{i_{r−1}i_{r}},

where i_{1} = i, i_{r} = j and i = i_{1} < i_{2} < i_{3} < ⋯ < i_{r−1} ≤ n.

Proof. This is by induction on n ≥ 2. □

For an n × n matrix A and 1 ≤ i_{1}, …, i_{r}, j_{1}, …, j_{r} ≤ n we let A([i_{1}, …, i_{r}], [j_{1}, …, j_{r}]) denote the determinant of the r×r submatrix of A where we use only the rows i_{1}, …, i_{r} and columns j_{1}, …, j_{r} from the matrix A. If [i_{1}, …, i_{r}] = [j_{1}, …, j_{r}] then we use the notation A[i_{1}, …, i_{r}]. We also let A_{[i_{1},...,i_{r}],[j_{1},...,j_{r}]} denote the determinant of the (n − r) × (n − r) submatrix of A where we use only the rows numbered {1, …, n} \ {i_{1}, …, i_{r}} and columns numbered {1, …, n} \ {j_{1}, …, j_{r}}. If [i_{1}, …, i_{r}] = [j_{1}, …, j_{r}], then we use the notation A_{[i_{1},...,i_{r}]}.

A key observation that simplifies many calculations is:

Proposition 3.2. For all n ≥ 2 we have det(Π′_{n} + I_{n}) = det(U_{n}) where U_{n} is as defined in Theorem 1. In fact all of the determinantal ideals determined by Π′_{n} + I_{n} and U_{n} are the same.

Proof. As we noted in the proof of [H2, Theorem 2.8], for any λ ∈ R the matrix Π_{n} − λI_{n} can be row-reduced to the following matrix (using only elementary matrices of determinant 1), which thus has the same determinantal ideals as Π_{n} − λI_{n}:

V_{n} = ( \begin{matrix}1 − λ & λa_{12} & … & λa_{1i} & … & λa_{1,n−1} & λa_{1n} \\ a_{21} & 1 − λ & … & λa_{2i} & … & λa_{2,n−1} & λa_{2n} \\ ⋮ & ⋮ & ... & ⋮ & & ⋮ & ⋮ \\ a_{i1} & a_{i2} & … & 1 − λ & … & λa_{i,n−1} & λa_{in} \\ ⋮ & ⋮ & & ⋮ & ... & ⋮ & ⋮ \\ a_{n−1,1} & a_{n−1,2} & … & a_{n−1,i} & … & 1 − λ & λa_{n−1,n} \\ a_{n1} & a_{n2} & … & a_{ni} & … & a_{n,n−1} & 1 − λ \end{matrix} )  (3.1.)

Putting λ = −1 and a_{ij} = −a_{ji} for i < j we obtain the matrix U_{n}. □

Proposition 3.3. Suppose that 2 is invertible in R. The matrix Π_{n} + I_{n} has rank 1 if and only if we have

a_{in}a_{ni} = −4, and 2a_{ij} = a_{in}a_{nj}, 2a_{ji} = −a_{jn}a_{ni}

for all 1 ≤ i < j < n. Further, the point where a_{ij} = 2 for i > j and a_{ij} = −2 for i < j satisfies the above conditions.

Proof. The last statement follows easily from the first. The basic fact here is that a matrix has rank 1 if and only if all of its 2 × 2 minors are zero. By Proposition 3.2 it suffices to consider the matrix V_{n} defined by (3.1) with λ = −1 . In this case one easily sees that for i < j < n we have

V_{n}([i, j]) = 4 + a_{ij}a_{ji}, and V_{n}([i, n], [j, n]) = 2a_{ij} − a_{in}a_{nj}.

Now for example if i < j < k < m ≤ n, then we have V_{n}([i, j], [k, m]) = a_{ik}a_{jm} − a_{im}a_{jk}, and modulo the ideal generated by the above relations this is equal to

\frac{1}{4}(a_{in}a_{nk}a_{jn}a_{nm} − a_{in}a_{nm}a_{jn}a_{nk}) = 0.

All other cases are similar. The result follows. □

In the antisymmetric case a_{ij} = −a_{ji} we see that there are only 2^{n−1} real solutions to these equations. Each such solution gives a coordinate vector (a_{21}, a_{23}, …, a_{n n−1}) where each entry is ±2. Let S_{n} denote the 2^{n−1} points determined in Proposition 3.3 at which Π′_{n} + I_{n} has rank 1. Note that the point where a_{ij} = 2, a_{ji} = −2 for 1 ≤ i > j ≤ n is in S_{n}.

Lemma 3.4. Suppose that 2 is invertible in R. Then there are exactly 2^{n−1} points where Π′_{n} + I_{n} has rank 1 and the braid group B_{n} acts transitively on them.

Proof. We have already noted that there are 2^{n−1} such points and that each point in S_{n} is completely determined by the n−1 coordinates corresponding to the variables a_{ni}, 1 ≤ i < n.

Thus we will denote each point of S_{n} by a list of n − 1 integers a_{n1}, a_{n2}, …, a_{n n−1} each of which is in {±2}. Thus the B_{n}-action on these n − 1 coordinates can easily be calculated (for the generators σ_{i}, i < n) using (1.2) as follows: for 1 ≤ i < n − 1 we have

σ_{i}(a_{ni}) = −a_{n i+1}; σ_{i}(a_{n i+1}) = a_{ni}; σ_{i}(a_{nj}) = a_{nj} j ≠ i, i + 1.  (3.2)

The action of σ_{n−1} is:

σ_{n−1}(a_{ni}) = \frac{a_{n n−1}a_{ni}}{2} for i ≤ n − 2 and σ_{n−1}(a_{n n−1}) = a_{n−1 n} = −a_{n n−1}. (3.3)

We now show that each point of S_{n} is in the orbit of π_{2} = (2, 2, …, 2). Suppose that p = (p_{1}, …, p_{n−1}) ∈ S_{n}. If p ≠ π_{2}, then there is 1 ≤ i ≤ n such that p_{1} = ⋯ = p_{i−1} = 2, p_{i} = −2. Define μ(p) = i. If μ(p) > 1, then using the above action and the relations satisfied by the a_{ij} given in Proposition 3.2 one checks that μ(σ_{i−1}(p)) > μ(p). Continuing we see that p is in the orbit of π_{2}.

If μ(p) = 1, then either p = −π_{2} or there is a unique j > 1 such that p_{1} = p_{2} = ⋯ = p_{j−1} = −2, p_{j} = 2. Let ν(p) = j. If p ≠ −π_{2}, then again one checks that ν(σ_{j−1}(p)) > ν(p) and so in each case we see that p is in the orbit of −π_{2}. But we now have σ_{1}(−π_{2}) = (2, −2, −2, …, −2), which puts us back into the situation where μ(σ_{1}(−π_{2})) = 2 > 1. □

Let W (D_{n}) < W (B_{n}) denote the Weyl or Coxeter groups of types B_{n}, D_{n} [GB, §5.3]. Thus W (B_{n}) is the group of all “signed permutations” of {±1, …, ±n} and W (D_{n}) is the subgroup where the product of the signs equals the sign of the permutation. As we pointed out in §1, the symmetric group S_{n}, which is the Coxeter group of type A_{n}, is a quotient of B_{n}. The above result is connected to the following result which shows that W (D_{n}) is also a quotient of B_{n}:

Proposition 3.5. For all n ≥ 2 there is an epimorphism B_{n} → W (D_{n}). Let κ_{i} denote the image of σ_{i}. Then the following is a presentation of W (D_{n}) with these generators:

< κ_{1}, …, κ_{n−1}|κ_{i}κ_{i+1}κ_{i} = κ_{i+1}κ_{i}κ_{i+1}, κ_{i}^{2} = κ_{i+1}κ_{i}^{2}κ_{i+1}, for i < n,
κ_{i}^{4} = 1, κ_{i}κ_{j} = κ_{j}κ_{i} for |i − j| > 1 > .

The group with the above presentation is isomorphic to the image of the action of B_{n} on the points of S_{n} if 2 is invertible in R.

Proof. For 1 ≤ i ≤ n we let τ_{i} = (i i + 1) ∈ S_{n} ⊂ W (B_{n}) denote the transposition and let ϵ_{i} ∈ W (B_{n}) denote multiplication of i by −1. Then one has the standard relations between these generators of W (D_{n}) and one checks that the elements κ_{i} = ϵ_{i}τ_{i} ∈ W (D_{n}) satisfy the braid relations and so a homomorphism B_{n} → W (D_{n}) is determined by σ_{i} ↦ κ_{i}.

To see that we have an epimorphism we note that κ_{i}^{2} = ϵ_{i}ϵ_{i+1} and that the subgroup < κ_{1}^{2}, …, κ_{n−1}^{2} > of W (D_{n}) is equal to W (D_{n})∩ < ϵ_{1}, …, ϵ_{n} > and has order 2^{n−1}. Further there is an epimorphism < κ_{1}, …, κ_{n−1} >→ S_{n} which kills < κ_{1}^{2}, …, κ_{n−1}^{2} >. Thus the order of < κ_{1}, …, κ_{n−1} > is at least n!2^{n−1} = |W (D_{n})|. However as < κ_{1}, …, κ_{n−1} > is a subgroup of W (D_{n}) it must be equal to it. Thus the homomorphism B_{n} → W (D_{n}) is onto.

Now we consider the given presentation. The relations show that < κ_{1}^{2}, …, κ_{n−1}^{2} > is a normal abelian subgroup of the group with the given presentation and that it has order 2^{n−1}. One further sees that the quotient by this normal subgroup is isomorphic to S_{n}. Since the relations given are easily shown to be satisfied by the images of σ_{i} under the homomorphism B_{n} → W (D_{n}) we see that the given presentation is a presentation for W (D_{n}).

Now using the action of the generators σ_{i} on the elements of S_{n} (see (3.2) and (3.3)) one can show that these generators satisfy the same relations as do the κ_{i}. One can check for example that the action of σ_{i}^{2} on S_{n} is to multiply the ith and i + 1th coordinates of elements of S_{n} by −1, showing that the images of σ_{1}^{2}, …, σ_{n−1}^{2} generate a normal abelian subgroup of order 2^{n−1}. The rest follows easily. □

Remark 3.6. The set of singular points S_{3} was considered by Goldman [Go, §6.1 case (d)] and corresponded to characters of representations into the centre of SL(2, ℝ).

§4 The case n = 3

Before embarking upon a general analysis of the rank 2 case we look at the situation where n = 3 and R = ℝ . Here we have a single generator for the ring of invariants, namely

c′_{31} = a_{21}^{2} + a_{31}^{2} + a_{32}^{2} − a_{21}a_{32}a_{31} + 3.

Now det(U_{3}) = −2(c_{31} + 1) and solving c′_{31} + 1 = 0 for (say) a_{21} we get two solutions:

a_{21}^{±} = \frac{a_{31}a_{32} ± \sqrt{(a_{31}^{2} − 4)(a_{32}^{2} − 4)}}{2}.

We will thus think of each a_{21}^{±} as being a function of a_{31}, a_{32}. Now as we are looking for real solutions we see that we need (a_{31}^{2} − 4)(a_{32}^{2} − 4) ≥ 0. This defines a domain E ⊂ ℝ^{2} for each a_{21}^{±}. The interior of E consists of 5 components, one of which has closure the region S determined by −2 ≤ a_{31}, a_{32} ≤ 2. Thus the solution set of c′_{31} + 1 = 0 over S consists of two smooth topological discs D_{−}, D_{+} in ℝ^{3} which meet along the image of the subset of E where a_{21}^{+} = a_{21}^{−} i.e. at all points where (a_{31}^{2} − 4)(a_{32}^{2} − 4) = 0. But this set is just the boundary of S. We will use coordinates for ℝ^{3} in the order (a_{21}, a_{31}, a_{32}). Then the boundary of S is a union of 4 straight line segments where a_{31}, a_{32} = ±2. One checks that if a_{31} = 2, then a_{21} = a_{32} and so we get a straight line segment from (2, 2, 2) to (−2, 2, −2) in this case. The other cases are similar and we get line segments connecting the points (2, 2, 2), (−2, 2, −2), (2, −2, −2), (−2, −2, 2) of S_{3}. These line segments give the 1-skeleton of a regular tetrahedron in ℝ^{3}. It follows from the above formula for a_{21}^{±} that the discs D_{−}, D_{+} meet along a piecewise-linear curve. Let D = D_{−} ∪ D_{+}.

Now one can check that the line segment between (2, 2, 2) and (2, −2, −2) is in D, and that similarly the line segment between (−2, 2, −2) and (−2, −2, 2) is in D. Further the smooth nature of the equations defining D_{±} shows that all the points on these two lines are smooth (except possibly at the end points). Clearly every point of the interior of D_{−} and the interior of D_{+} is a smooth point. Thus the only possibility of a non-smooth point is a point of the piecewise linear curve D_{−} ∩ D_{+}. However we can also do the above analysis by solving for a_{31} or a_{32} (instead of a_{21}). Doing so, and repeating the above argument, shows that in fact the only possible singular points are at the four points (2, 2, 2), (−2, 2, −2), (2, −2, −2), (−2, −2, 2).

One can also check that D meets the 3-ball [−2, 2]^{3} only at the six line segments joining the points (2, 2, 2), (−2, 2, −2), (2, −2, −2), (−2, −2, 2).

Now B_{3} acts on the set V′_{31} of real solutions of c′_{31} +1 = 0. The projection of V′_{31} onto the a_{31}, a_{32}-axis gives a region consisting of S together with four regions Q_{1}, …, Q_{4} determined by |a_{31}|, |a_{32}| ≥ 2. Each of these four regions is a plane meeting S at a single point. The part of V′_{31} above each Q_{i} is thus an open disc which meets D at one of the four points (2, 2, 2), (−2, 2, −2), (2, −2, −2), (−2, −2, 2). Again one easily sees that it is smooth except at these points. One gets a very nice picture of V′_{31} upon drawing it using something like maple. One clearly sees that D has similarities with the standard tetrahedron embedded in [−2, 2]^{3} ⊂ ℝ^{3}. See Figure 2 for a graphical representation of D. A similar drawing may be found in [Go, Fig. 2].

Figure 2.

Now D bounds a 3-ball B ⊂ ℝ^{3} and the action of B_{3} clearly fixes B since B_{3} fixes the origin and D. Further for all 0 < t < 4 we consider the set of solutions to

a_{21}a_{31}a_{32} − a_{21}^{2} − a_{31}^{2} − a_{32}^{2} + 4 − t = 0  (4.1)

in ℝ^{3}. This equation occurs frequently in the literature; see for example [Go, K1]. As in the above we can solve for a_{21} to get:

a_{21}^{±}(t) = \frac{a_{31}a_{32} ± \sqrt{(a_{31}^{2} − 4)(a_{32}^{2} − 4) − 4t}}{2}.

Thus for 0 < t < 4 we see that a part of the solution set is a smooth 2-sphere D^{(t)} ⊂ B. We have D = D^{(0)}. The fact that this 2-sphere is smooth for t ∈ (0, 4) is seen by looking at the Jacobian of the equation (4.1) and noting that the only singular points are at the four points (2, 2, 2), (−2, 2, −2), (2, −2, −2), (−2, −2, 2), together with (0, 0, 0).

From [H1] we know that the representation B_{3} → Aut(ℚ′_{3}) has kernel equal to its centre Z(B_{3}) =< (σ_{1}σ_{2})^{3} >≅ ℤ. Since B is a 3-ball there are points (b_{21}, b_{31}, b_{32}) ∈ B for which the real numbers b_{21}, b_{31}, b_{32} are algebraically independent. Thus the action of B_{3}/Z(B_{3}) at these points is faithful. Now the set of points (b_{21}, b_{31}, b_{32}) ∈ B with algebraically independent coordinates is dense in the ball B. Thus there are a dense set of values of t ∈ (0, 4) such that the action of B_{3}/Z(B_{3}) is faithful on the spheres D^{(t)}.

We now consider the question of convexity. This we will prove by showing that the Hessians H_{4}^{±} of the function a_{21}^{±} = a_{31}a_{32} ± \sqrt{(a_{31}^{2} − 4)(a_{32}^{2} − 4)}/2 are positive and negative definite (except on a set of measure 0). We will do H_{4}^{−}, showing that it is positive definite, the other case being similar. This Hessian is

H_{4}^{−} = ( \begin{matrix}2 \frac{a_{32}^{2}−4}{(a_{31}^{2}−4)\sqrt{(a_{32}^{2}−4)(a_{31}^{2}−4)}} & −1/2 \frac{a_{31}a_{32}−\sqrt{(a_{32}^{2}−4)(a_{31}^{2}−4)}}{\sqrt{(a_{32}^{2}−4)(a_{31}^{2}−4)}} \\ −1/2 \frac{a_{31}a_{32}−\sqrt{(a_{32}^{2}−4)(a_{31}^{2}−4)}}{\sqrt{(a_{32}^{2}−4)(a_{31}^{2}−4)}} & 2 \frac{a_{31}^{2}−4}{(a_{32}^{2}−4)\sqrt{(a_{32}^{2}−4)(a_{31}^{2}−4)}} \end{matrix} ) .

Now since −2 ≤ a_{31}, a_{32} ≤ 2 we see that the diagonal entries are positive on the interior of the domain. Thus we need only show that det(H_{4}^{−}) is positive. Now we have

det(H_{4}^{−}) = −1/2 \frac{a_{31}^{2}a_{32}^{2} − a_{31}a_{32}\sqrt{(a_{32}^{2} − 4) (a_{31}^{2} − 4)} − 2 a_{31}^{2} − 2 a_{32}^{2}}{(a_{32}^{2} − 4) (a_{31}^{2} − 4)} .

Solving det(H_{4}^{−}) = 0 gives the solutions a_{31} = ±a_{32}. Call this set Z and notice that [−2, 2]^{2} \ Z has components on the interior of each of which det(H_{4}) is positive.

We summarise these results as follows:

Theorem 4.1. The quotient B_{3}/Z(B_{3}) ≅ P SL_{2}(ℤ) acts faithfully on a convex topological ball B ⊂ [−2, 2]^{3} ⊂ ℝ^{3} with boundary D containing the four points

(2, 2, 2), (−2, 2, −2), (2, −2, −2), (−2, −2, 2).

The 2-sphere D is smooth except at these points. The six line segments connecting these points are also in D and form the 1-skeleton of a regular tetrahedron in ℝ^{3}. Further, D meets the 3-ball [−2, 2]^{3} only at these six line segments.

Moreover, for all 0 < t < 4 there is a component D^{(t)} of the solution set of

a_{21}a_{31}a_{32} − a_{21}^{2} − a_{31}^{2} − a_{32}^{2} + 4 − t = 0

that is a B_{3}/Z(B_{3})-invariant smooth 2-sphere inside B. There is a dense set of values of such t for which this action of B_{3}/Z(B_{3}) on D^{(t)} is faithful. □

§5 Results on symmetric matrices

Proposition 5.1. Let U = (u_{ij}) be any symmetric n×n matrix. Then for all 1 ≤ j < i ≤ n the determinant det(U) is quadratic in u_{ij} and its discriminant relative to u_{ij} is equal to 4U_{[i]}U_{[j]}.

In particular, if U = Π′_{n} + I_{n}, then for all 1 ≤ j < i ≤ n the determinant det(Π′_{n} + I_{n}) is quadratic in a_{ij} and its discriminant relative to a_{ij} is equal to 4U_{[i]}U_{[j]}.

Proof. The second statement will follow from the first, which we now prove. Since there is an action of S_{n} on the entries of U which does not change the determinants we need only prove the result for i = 2, j = 1. Since u_{21} only occurs in the (1, 2) and (2, 1) positions in U, det(U) is clearly quadratic in u_{21}. Now from [M; p. 370] we see that

U_{[1]}U_{[2]} − (U_{[1],[2]})^{2} = det(U)U_{[1,2]}.  (5.1)

Now

U_{[1],[2]} = u_{21}U_{[1,2],[1,2]} − u_{31}U_{[1,2],[1,3]} + u_{41}U_{[1,2],[1,4]} − ⋯ = u_{21}U_{[1,2],[1,2]} + X.

Since U_{[1]}, U_{[2]} and U_{[1,2],[1,2]} are constant relative to u_{21} it follows that the discriminant of

−U_{[1]}U_{[2]} + (U_{[1],[2]})^{2} = −U_{[1]}U_{[2]} + (u_{21}U_{[1,2]} + X)^{2}
= u_{21}^{2}(U_{[1,2]})^{2} + 2XU_{[1,2]}u_{21} − U_{[1]}U_{[2]} + X^{2}

relative to the variable u_{21} is

4X^{2}(U_{[1,2]})^{2} − 4(U_{[1,2]})^{2}(X^{2} − U_{[1]}U_{[2]}) = 4(U_{[1,2]})^{2}U_{[1]}U_{[2]}.

Now the appearance of the extra factor U_{[1,2]} on the right hand side of (5.1) shows that the discriminant of det(U) relative to u_{21} is 4U_{[1]}U_{[2]}. □

Proposition 5.2. Let 1 < r ≤ n, r ∈ 2ℤ and let R be a ring of characteristic 2. Then there is a symmetric matrix of rank r over R′_{n} with 0s on the diagonal, all of whose off-diagonal entries are non-zero.

Proof. Let m = n − r and let Q_{r} = (a_{ij}) be a symmetric matrix with 0s on the diagonal. Let B = (b_{ij}) be a generic m × r matrix where b_{ij} = a_{i+r j}. Note that Q_{r} has rank r since r is even. Also define n × n matrices

M_{n} = ( \begin{matrix}Q_{r} & 0 \\ 0 & 0 \end{matrix} ) , E_{n} = ( \begin{matrix}I_{r} & 0 \\ B & I_{m} \end{matrix} ) .

Clearly M_{n} has rank r. We claim that E_{n}M_{n}E_{n}^{t} satisfies the requirements of Proposition 5.2. Now we note that E_{n}M_{n}E_{n}^{t} is symmetric; further we have

E_{n}M_{n}E_{n}^{t} = ( \begin{matrix}I_{r} & 0 \\ B & I_{m} \end{matrix} ) ( \begin{matrix}Q_{r} & 0 \\ 0 & 0 \end{matrix} ) ( \begin{matrix}I_{r} & B^{t} \\ 0 & I_{m} \end{matrix} )

= ( \begin{matrix}Q_{r} & 0 \\ BQ_{r} & 0 \end{matrix} ) ( \begin{matrix}I_{r} & B^{t} \\ 0 & I_{m} \end{matrix} ) = ( \begin{matrix}Q_{r} & Q_{r}B^{t} \\ BQ_{r} & BQ_{r}B^{t} \end{matrix} ) .

The ps entry of BQ_{r} is ∑_{i=1}^{r} b_{pi}a_{is} ≠ 0 and the pt entry of BQ_{r}B^{t} is ∑_{s=1}^{r} ∑_{i=1}^{r} b_{pi}a_{is}b_{ts}. It follows that all the off-diagonal entries of E_{n}M_{n}E_{n}^{t} are non-zero and that all the diagonal entries are zero. □

§6 The n = 4 case

The case n = 4 has some features that do not seem to arise in the n > 4 cases, so we look at these in this section. The reason for the exceptional behaviour is the existence of the well-known epimorphism B_{4} → B_{3}, (σ_{1} ↦ σ_{1}, σ_{2} ↦ σ_{2}, σ_{3} ↦ σ_{1}). The most notable difference is the existence of a pair of subspaces that are permuted by B_{4}:

Theorem 6.1. For n = 4 the braid group B_{4} acts on a disjoint union S^{2+} ∪ S^{2−} of two 2-spheres. The union of these spheres contain the singular points S_{4}. For each ϵ = ± there is a smooth embedding ι_{ϵ} : ℝ^{3} → ℝ^{6} such that the images P_{ϵ} of these maps are permuted by B_{4} and contain the two 2-spheres i.e. S^{2±} ⊂ P_{±}. We have P_{−} ∩ P_{+} = {0}.

Proof. We define the embedded planes relative to the coordinates (a_{21}, a_{31}, a_{32}, a_{41}, a_{42}, a_{43}) as follows:

P_{+} = {(a, b, c, c, d, a) ∈ ℝ^{6}|b + d = ac};
P_{−} = {(a, b, c, −c, −d, −a) ∈ ℝ^{6}|b + d = ac}.

It is easy to check that σ_{i}(P_{ϵ}) = P_{−ϵ} for all 1 ≤ i < 4 and ϵ = ±; for example if (a, b, c, c, d, a) ∈ P_{+}, then

σ_{1}(a, b, c, c, d, a) = (−a, c − ab, b, d − ac, c, a) = (a′, b′, c′, −c′, −d′, −a′),

where b′ + d′ = (c − ab) + (−c) = −ab = a′c′ and so σ_{1}(a, b, c, c, d, a) ∈ P_{−}.

One now also checks that each point in S_{4} is contained in one of the P_{±}. Next note that if we require that the a, b, c, d in the above also satisfy

abc − a^{2} − b^{2} − c^{2} + 4 = 0,

then we obtain the 2-spheres S^{2+} ⊂ P_{+}, S^{2−} ⊂ P_{−}. That these are 2-spheres follows from the arguments of §3. These are contained in the same level set of the invariants c′_{41}, c′_{42} as the points of S_{4}. One easily checks that P_{−} ∩ P_{+} = {0} and so that S^{2+}, S^{2−} are disjoint. Next one checks that the union of these spheres is also fixed by the action of B_{4}. To do this one only needs to check the action of the generators σ_{i}, i = 1, 2, 3 and this is easy. □

§7 The rank 2 case

In this section we assume that R = ℝ and we investigate the nature of the subset V_{n2} ⊂ ℝ^{\binom{n}{2}} where the rank of the matrix Π′_{n} + I_{n} is 2. This set is invariant under the action of B_{n} by Proposition 3.2.

Let λ = (λ_{21}, λ_{31}, λ_{32}, …, λ_{nn−1}) ∈ ℝ^{\binom{n}{2}}. Then λ is called a tetrahedral point if for all 1 ≤ i < j < k ≤ n we have

λ_{kj}λ_{ki}λ_{ji} − λ_{kj}^{2} − λ_{ki}^{2} − λ_{ji}^{2} + 4 = 0.

Lemma 7.1. Let λ = (λ_{21}, λ_{31}, …, λ_{nn−1}) ∈ ℝ^{\binom{n}{2}} be a point at which rank(Π′_{n} + I_{n}) is at most 2. Then λ is a tetrahedral point.

Proof. Here we need only consider the minors U_{n}([i, j, k]); these give exactly the tetrahedral condition. □

Lemma 7.2. The only singular points on V_{42} are the points in the set S_{4}.

Proof. For any square matrix M the singular points of the determinantal ideal of all 3 × 3 minors is given by the determinantal ideal of all 2×2 minors [ACGH, p. 69]. This is exactly the set S_{n} (see Lemma 3.4). □.

Remark 7.3. We have seen above that the set S_{n} is fixed by the action of B_{n}, however these are not the only finite orbits: For n = 3 the orbit of (\sqrt{2}, 0, −1) has 36 elements and the action of B_{3} gives a group of order 2654208 = 2^{15}3^{4}. The orbit of (−\sqrt{2 + \sqrt{2}}, −\sqrt{2 + \sqrt{2}}, 2) has 96 elements and the action of B_{3} gives a group of order 1536 = 2^{9}3. The B_{4}-orbit of the tetrahedral point (\sqrt{3}, −\sqrt{3}, −1, 0, 1, 1) ∈ V_{42} consists of 288 points; the action of B_{4} on these points gives a group of order 165888 = 2^{11}3^{4}.

Now for 1 ≤ i < j < k ≤ n there is a projection π_{ijk} : ℝ^{\binom{n}{2}} → ℝ^{3} which forgets those coordinates a_{rs} with r, s ∉ {i, j, k}. The fact that points v ∈ V_{n2} satisfy the tetrahedral condition shows that the image of each such π_{ijk} consists of points (x, y, z) ∈ ℝ^{3} such that xyz − x^{2} − y^{2} − z^{2} + 4 = 0. The nature of this solution set was investigated in §3 from which it is natural to define:

V_{n2}^{(2)} = V_{n2} ∩ [−2, 2]^{\binom{n}{2}}.

Note that V_{n2}^{(2)} is a compact set and is invariant under the action of B_{n}.

Now we solve the tetrahedral equations. For any 1 ≤ j < i < n the coordinates a_{ij}, a_{ni}, a_{nj} satisfy the tetrahedral relation and so we can solve for a_{ij} as a function of a_{ni}, a_{nj}:

a_{ij} = a_{ij}^{ϵ_{ij}}(a_{n1}, a_{n2}, …, a_{n n−1}) = a_{nj}a_{ni}/2 + ϵ_{ij}\sqrt{(a_{nj}^{2} − 4)(a_{ni}^{2} − 4)}/2,  (7.1)

where ϵ_{ij} = ϵ_{ji} = ±1. For convenience we also put a_{ni}^{±1}(a_{n1}, a_{n2}, …, a_{n n−1}) = a_{ni}.

Lemma 7.4. For any z = (a_{n1}, a_{n2}, …, a_{n n−1}) ∈ [−2, 2]^{n−1} and ϵ = (ϵ_{21}, ϵ_{31}, …, ϵ_{n n−1}) ∈ \{±1\}^{\binom{n}{2}} the point f_{ϵ}(z) = (a_{21}^{ϵ_{21}}(z), a_{31}^{ϵ_{31}}(z), …, a_{n n−1}^{ϵ_{n n−1}}(z)) ∈ ℝ^{\binom{n}{2}} with the a_{ij}^{ϵ_{ij}}, 1 ≤ j < i < n, given by (7.1) is in V_{n2} if and only if the following conditions are satisfied:

ϵ_{ij}ϵ_{jk}ϵ_{ik} = −1 for all distinct 1 ≤ i, j, k < n.  (7.2)

Proof. To prove the Lemma we need to show that any (a_{21}^{ϵ_{21}}, a_{31}^{ϵ_{31}}, …, a_{n n−1}^{ϵ_{n n−1}}) gives a point in R^{\binom{n}{2}} (with the a_{ij}^{ϵ_{ij}}, 1 ≤ j < i < n given by (7.1)) satisfying all of the equations U_{n}([i_{1}, i_{2}, i_{3}], [j_{1}, j_{2}, j_{3}]) = 0. Now the symmetric group acts naturally on U_{n} by permutation of subscripts and the subgroup S_{n−1} fixes any point where the ϵ_{ij} are given by (7.2). Thus this S_{n−1} acts on the U_{n}([i_{1}, i_{2}, i_{3}], [j_{1}, j_{2}, j_{3}]) and we need only check U_{n}([i_{1}, i_{2}, i_{3}], [j_{1}, j_{2}, j_{3}]) = 0 for U_{n}([1, 2, 3], [j_{1}, j_{2}, j_{3}]) and U_{n}([1, 2, n], [j_{1}, j_{2}, j_{3}]). In fact even here there are only a small number of cases to check depending on the cardinality of the set {1, 2, 3} ∩ {j_{1}, j_{2}, j_{3}} or {1, 2, n} ∩ {j_{1}, j_{2}, j_{3}} (respectively). For example we have (using ϵ_{ij}^{2} = 1)

U_{n}([1, 2, 3]) = (a_{n1}^{2} − 4)(a_{n2}^{2} − 4)(a_{n3}^{2} − 4)(ϵ_{12}ϵ_{13}ϵ_{23} + 1)/8
+ a_{n2}a_{n3}\sqrt{(a_{n2}^{2} − 4)(a_{n3}^{2} − 4)}(a_{n1}^{2} − 4)(ϵ_{23} + ϵ_{12}ϵ_{13})/8
+ a_{n1}a_{n3}\sqrt{(a_{n1}^{2} − 4)(a_{n3}^{2} − 4)}(a_{n2}^{2} − 4)(ϵ_{13} + ϵ_{23}ϵ_{12})/8
+ a_{n1}a_{n2}\sqrt{(a_{n1}^{2} − 4)(a_{n2}^{2} − 4)}(a_{n3}^{2} − 4)(ϵ_{12} + ϵ_{13}ϵ_{23})/8.

This is zero if and only if ϵ_{12}ϵ_{13}ϵ_{23} = −1 and so we get ϵ_{ij}ϵ_{ik}ϵ_{jk} = −1 for all distinct 1 ≤ i, j, k < n. Now one similarly checks that with these conditions satisfied we also have U_{n}([1, 2, 3], [1, 2, 4] = 0, U_{n}([1, 2, 3], [3, 4, 5] = 0, U_{n}([1, 2, 3], [4, 5, 6] = 0, U_{n}([1, 2, n], [1, 2, 3] = 0 etc. For example U_{n}([1, 2, 3], [1, 2, 4] can be written as:

\frac{1}{8}\sqrt{(a_{n1}^{2} − 4)(a_{n3}^{2} − 4)}a_{n1}a_{n4}(a_{n2}^{2} − 4)(ϵ_{13} + ϵ_{12}ϵ_{23})

+ \frac{1}{8}\sqrt{(a_{n2}^{2} − 4)(a_{n3}^{2} − 4)}a_{n2}a_{n4}(a_{n1}^{2} − 4)(ϵ_{23} + ϵ_{12}ϵ_{13})

+ \frac{1}{8}\sqrt{(a_{n1}^{2} − 4)(a_{n4}^{2} − 4)}a_{n1}a_{n3}(a_{n2}^{2} − 4)(ϵ_{14} + ϵ_{12}ϵ_{24})

+ \frac{1}{8}\sqrt{(a_{n2}^{2} − 4)(a_{n4}^{2} − 4)}a_{n2}a_{n3}(a_{n1}^{2} − 4)(ϵ_{24} + ϵ_{12}ϵ_{14})

− \frac{1}{8}\sqrt{(a_{n1}^{2} − 4)(a_{n2}^{2} − 4)(a_{n3}^{2} − 4)(a_{n4}^{2} − 4)}a_{n1}a_{n2}(2ϵ_{12}ϵ_{34} − ϵ_{14}ϵ_{23} − ϵ_{13}ϵ_{24})

+ \frac{1}{8}\sqrt{(a_{n3}^{2} − 4)(a_{n4}^{2} − 4)}[16ϵ_{24}(ϵ_{23} + ϵ_{12}ϵ_{13}) + 16ϵ_{23}(ϵ_{24} + ϵ_{12}ϵ_{14})

+ a_{51}^{2}a_{52}^{2}(ϵ_{12}ϵ_{13}ϵ_{24} + ϵ_{12}ϵ_{23}ϵ_{14} − 2ϵ_{34}) − 4a_{n1}^{2}(ϵ_{13}ϵ_{14} + ϵ_{12}ϵ_{14}ϵ_{23} + ϵ_{12}ϵ_{13}ϵ_{24} − ϵ_{34})

− 4a_{n2}^{2}(ϵ_{23}ϵ_{24} + ϵ_{12}ϵ_{23}ϵ_{14} + ϵ_{12}ϵ_{13}ϵ_{24} − ϵ_{34})].

The other cases are similar. This proves the result. □

Conjecture 7.5. Now if 1 < r ≤ n, then for 1 ≤ j < i ≤ n − r + 1 we can solve the quadratic equation U_{n}[i, j, n + 2 − r, n + 3 − r, …, n] = 0 for a_{ij} to get a solution depending on a parameter ϵ_{ij} = ±1. We conjecture that doing so for all such i, j will give a matrix of rank r if and only if we have ϵ_{ij}ϵ_{jk}ϵ_{ik} = (−1)^{r+1}. What we have proved above is that this is true for r = 2. As indicated in the above proof each case requires only a finite amount of checking and we have also checked that the conjecture holds for r = 3, 4.

Returning to the case r = 2 one can check that the conditions (7.2) are satisfied if we have

ϵ_{ij} = −ϵ_{1i}ϵ_{1j}

for all 1 < i < j < n. Thus there are 2^{n−2} choices corresponding to the values of ϵ_{21}, ϵ_{31}, …, ϵ_{n−1,1} and we will now let ϵ = (ϵ_{21}, ϵ_{31}, …, ϵ_{n−1,1}). Thus by (7.1) for each ϵ satisfying (7.2) we have a function

f_{ϵ} = f_{ϵ}(a_{n1}, …, a_{nn−1}) : [ − 2, 2]^{n−1} → ℝ^{\binom{n}{2}}.

The image of [−2, 2]^{n−1} is an (n − 1) − ball in ℝ^{\binom{n}{2}}. We will call it an ϵ − cube.

These ϵ-cubes meet along only their faces and we now describe these identifications and a certain fundamental groupoid. Let p_{ϵ} = f_{ϵ}(0, 0, …, 0). This is the centre of the ϵ-cube. By a face of the ϵ-cube we will mean the subset determined by a_{ni} = ±2 for some 1 ≤ i < n.

For 1 ≤ i < n and μ ∈ {±1} we let F_{i,ϵ,μ} be the face of the ϵ-cube determined by a_{ni} = μ2. The following two conditions are checked using (7.1): (7.i) If ϵ′ differs from ϵ only in the ϵ_{1i} place (i > 1), then F_{i,ϵ,μ} is canonically identified with F_{i,ϵ′,μ}: f_{ϵ}(z) = f_{ϵ′}(z) for all z with a_{ni} = μ2. (7.ii) If ϵ′ = −ϵ, then F_{1,ϵ,μ} is similarly identified with F_{1,ϵ′,μ}.

These are the only ways that the faces are identified. Thus the faces are identified in pairs. For example, when n = 4 the ϵ = (+, +) cube is as shown in Figure 3:

Figure 3

Here we have indicated the direction in which each of a_{41}, a_{42}, a_{43} increases and −2 ≤ a_{41}, a_{42}, a_{43} ≤ 2. Thus, for example, F_{1,(+,+),+1} is the face B_{++}B′_{++}C′_{++}C_{++}. There are three other such cubes whose vertices we label similarly and they have faces which are identified as follows:

ABB′A′_{++} ≡ ABB′A′_{+−}; DCC′D′_{++} ≡ DCC′D′_{+−}
ADD′A′_{++} ≡ ADD′A′_{−−}; BB′C′C_{++} ≡ BB′C′C_{−−}
ABCD_{++} ≡ ABCD_{−+}; A′B′C′D′_{++} ≡ A′B′C′D′_{−+};
AA′D′D_{+−} ≡ AA′D′D_{−+}; BB′C′C_{+−} ≡ BB′C′C_{−+};
ABCD_{+−} ≡ ABCD_{−−}; A′B′C′D′_{+−} ≡ A′B′C′D′_{−−};
ABB′A′_{−+} ≡ ABB′A′_{−−}; DCC′D′_{−+} ≡ DCC′D′_{−−}.

Now making the ABB′A′_{++} ≡ ABB′A′_{+−}, DCC′D′_{++} ≡ DCC′D′_{+−} identifications in the above list gives a solid torus, as does making the ABB′A′_{−+} ≡ ABB′A′_{−−}, DCC′D′_{−+} ≡ DCC′D′_{−−} identifications. Now the rest of the identifications give the way of identifying the boundaries of these two solid tori. See Figure 4:

Figure 4.

One checks that all of A_{++}, A_{+−}, A_{−+}, A_{−−} are identified and similarly for B, C, D and A′, B′, C′, D′. The edges AB_{++}, AB_{+−}, AB_{−+}, AB_{−−} are similarly all identified. One also checks that at each edge all four cubes come together as if they were stacked in ℝ^{3}. The resulting space is a manifold except at the 8 vertices (the points of S_{4}), where one can check that the link is an ℝℙ^{2}.

Now let T^{n−1} be the (n − 1)-torus S^{1} × ⋯ × S^{1} (n − 1 times). Here we represent S^{1} as ℝ mod 2π. Thus S^{1} = I_{1} ∪ I_{2} where I_{1} = [0, π], I_{2} = [π, 2π]. Let α : T^{n−1} → T^{n−1} be the antipodal map α(x) = −x for all x ∈ T^{n−1}. We will think of α as acting on all the T^{n−1}. It is clear that α respects adjacencies. Further, the fixed point set of α for this action is {0, π}^{n−1} and this set is contained in each cube.

Theorem 7.6. For all n > 3 the orbifold quotient T^{n−1}/ < α > is homeomorphic to V_{n2}^{(2)}. The 2^{n−1} singular points of V_{n2}^{(2)} are non-manifold points whose links are all homeomorphic to ℝℙ^{n−2}. The open manifold V_{n2}^{(2)} \ S_{n} is acted upon by B_{n}.

Proof. We will exhibit T^{n−1} as a cubical complex which is invariant under the map α and then show that the cubes in the quotient complex have the same face identifications as we obtained for V_{n2}^{(2)} above.

Now we naturally have

T^{n−1} = ⋃_{i_{1},i_{2},...,i_{n−1}∈{1,2}} I_{i_{1}} × I_{i_{2}} × ⋯ × I_{i_{n−1}}.

Note that there are 2^{n−1} such cubes in this decomposition and we may denote each cube by C(i_{1}, i_{2}, …, i_{n−1}). Now α(I_{1}) = I_{2}, α(I_{2}) = I_{1}. Thus in the quotient T^{n−1}/ < α > each such cube can be represented by a cube of the form I_{1} × …. Now two distinct cubes C(i_{1}, i_{2}, …, i_{n−1}) and C(i′_{1}, i′_{2}, …, i′_{n−1}) have a face in common if and only if the sequences (i_{1}, i_{2}, …, i_{n−1}) and (i′_{1}, i′_{2}, …, i′_{n−1}) differ in exactly one position.

To each cube C(i_{1}, i_{2}, …, i_{n−1}) we associate a sequence ϵ(i_{1}, i_{2}, …, i_{n−1}) = (ϵ_{2}, …, ϵ_{n−1}) as follows:

ϵ_{j} = \{ \begin{matrix}+1 if i_{j} = 1, i_{1} = 1; \\ −1 if i_{j} = 2, i_{1} = 1. \end{matrix} ϵ_{j} = \{ \begin{matrix}−1 if i_{j} = 1, i_{1} = 2; \\ +1 if i_{j} = 2, i_{1} = 2. \end{matrix}

Then C(i_{1}, i_{2}, …, i_{n−1}) ↦ Im(f_{ϵ(i_{1},i_{2},...,i_{n−1})}) gives a one-to-one correspondence between the classes of cubes in T^{n−1}/ < α > and the cubes Im(f_{ϵ}). We now show that if C(i_{1}, i_{2}, …, i_{n−1}) and C(i′_{1}, i′_{2}, …, i′_{n−1}) share a face, then so do Im(f_{ϵ(i_{1},i_{2},...,i_{n−1})}) and Im(f_{ϵ(i′_{1},i′_{2},...,i′_{n−1})}).

Now C(i_{1}, i_{2}, …, i_{n−1}) and C(i′_{1}, i′_{2}, …, i′_{n−1}) share a face if and only if the sequences (i_{1}, i_{2}, …, i_{n−1}) and (i′_{1}, i′_{2}, …, i′_{n−1}) differ in exactly one entry. First, if (i_{1}, i_{2}, …, i_{n−1}) and (i′_{1}, i′_{2}, …, i′_{n−1}) differ only in the jth entry where j > 1, then Im(f_{ϵ(i_{1},i_{2},...,i_{n−1})}) and Im(f_{ϵ(i′_{1},i′_{2},...,i′_{n−1})}) share a face by (7.i) above. Otherwise if (i_{1}, i_{2}, …, i_{n−1}) and (i′_{1}, i′_{2}, …, i′_{n−1}) differ only in the first entry, then Im(f_{ϵ(i_{1},i_{2},...,i_{n−1})}) and Im(f_{ϵ(i′_{1},i′_{2},...,i′_{n−1})}) share a face by (7.ii) above.

Lastly, the link of a vertex of one of the C(i_{1}, i_{2}, …, i_{n−1}) in T^{n−1} is an S^{n−2} and so the link of a vertex of Im(f_{ϵ(i_{1},i_{2},...,i_{n−1})}) in T^{n−1}/ < α > is an ℝℙ^{n−2}. □.

Returning to the n = 4 case we wish to explicitly find the fundamental group π_{1}(V_{42}^{(2)}). We will find π_{1}(V_{42}^{(2)}) by first finding the fundamental groupoid π_{1}(V_{42}^{(2)}, P), where P = ∪_{(±,±)}p_{±±}. For information on fundamental groupoids see [Br].

For 1 ≤ i < n, μ ∈ {±1} we let g_{i,ϵ,μ} be an arc from p_{ϵ} to the centre of F_{i,ϵ,μ}. Thus for i > 1 and ϵ, ϵ′ differing only in the ith entry we have an arc g_{i,ϵ,μ}g_{i,ϵ′,μ}^{−1} from p_{ϵ} to p_{ϵ′}. Similarly g_{1,ϵ,μ}g_{1,−ϵ,μ}^{−1} goes from p_{ϵ} to p_{−ϵ}.

Proposition 7.7. The fundamental groupoid π_{1}(V_{42}^{(2)}, P) has the following generators and relations:

< a_{1}, b_{1}, c_{1}, d_{1}, e_{1}, f_{1}, a_{2}, b_{2}, c_{2},d_{2}, e_{2}, f_{2}|
e_{1}b_{1}f_{1}a_{1}^{−1}, e_{1}b_{2}f_{1}a_{2}^{−1}, e_{2}b_{1}f_{2}a_{1}^{−1}, e_{2}b_{2}f_{2}a_{2}^{−1},
c_{1}b_{1}^{−1}d_{1}a_{1}^{−1}, c_{1}b_{2}^{−1}d_{1}a_{2}^{−1}, c_{2}b_{1}^{−1}d_{2}a_{1}^{−1}, c_{2}b_{2}^{−1}d_{2}a_{2}^{−1},
c_{1}f_{1}d_{1}^{−1}e_{1}^{−1}, c_{1}f_{2}d_{1}^{−1}e_{2}^{−1}, c_{2}f_{1}d_{2}^{−1}e_{1}^{−1}, c_{2}f_{2}d_{2}^{−1}e_{2}^{−1} > .

The fundamental group π_{1}(V_{42}^{(2)}) has the following generators and relations:

< b_{1}, f_{2}, b_{2}, d_{2}|b_{1}^{2}, f_{2}^{2}, b_{2}^{2}, d_{2}^{2}, (f_{2}b_{1}b_{2})^{2}, (d_{2}b_{1}b_{2})^{2}, (d_{2}b_{1}f_{2})^{2} > .

The group π_{1}(V_{42}^{(2)}) has a normal subgroup of index 2 which is isomorphic to ℤ^{3}.

Proof. First define some generators as follows:

a_{1} = g_{1+++}g_{1−−+}^{−1}; b_{1} = g_{1+−+}g_{1−++}^{−1}; c_{1} = g_{2+++}g_{21−++}^{−1};
d_{1} = g_{2+−+}g_{2−−+}^{−1}; e_{1} = g_{3+++}g_{3+−+}^{−1}; f_{1} = g_{3−++}g_{3−−+}^{−1};
a_{2} = g_{1++−}g_{1−−−}^{−1}; b_{2} = g_{1+−−}g_{1−+−}^{−1}; c_{2} = g_{2++−}g_{2−+−}^{−1};
d_{2} = g_{2+−−}g_{2−−−}^{−1}; e_{2} = g_{3++−}g_{3+−−}^{−1}; f_{2} = g_{3−+−}g_{3−−−}^{−1}.

These are arcs which connect the points of P. They correspond to the identifications of the faces given above; for example a_{1} corresponds to the identification BB′C′C_{++} ≡ BB′C′C_{−−}. They generate π_{1}(V_{42}^{(2)}, P). Now each edge of each cube determines a relation. The set of such generators and relations suffices for a presentation of π_{1}(V_{42}^{(2)}, P). Since each edge is adjacent to each of the four cubes we obtain relators of length 4 in the above generators. One calculates that they are as indicated in Proposition 7.7. For example the edge CC′ corresponds to the first relator given in the presentation.

To obtain a presentation for the fundamental group from the given presentation for the fundamental groupoid we just need to “collapse a maximal tree” in the generator graph; this collapsed tree will give the base point for the fundamental group. We choose the tree determined by a_{1}, c_{1}, e_{1}. A calculation now gives the presentation indicated in the proposition. The last statement follows from the given presentation for π_{1}(V_{42}^{(2)}), since one can easily show that the subgroup generated by f_{2}b_{1}, b_{2}b_{1}, d_{2}b_{1} is isomorphic to ℤ^{3} and has index 2. □

Now B_{4} acts on V_{42}^{(2)} and at any point of V_{42}^{(2)} the determinant of U_{4} is zero. This however leaves the possibility that not all of the invariants c′_{4i}, i = 1, …, 4 are constant on V_{42}^{(2)}. In fact one checks that

c′_{43} = −a_{21}a_{31}a_{32}+a_{21}a_{32}a_{41}a_{43} − a_{21}a_{41}a_{42} − a_{32}a_{42}a_{43} − a_{31}a_{41}a_{43}
+ a_{21}^{2} + a_{31}^{2} + a_{32}^{2} + a_{41}^{2} + a_{42}^{2} + a_{43}^{2} − 8

is not constant on V_{42}^{(2)}. Thus V_{42}^{(2)} is a union of the level sets of c′_{43}, each such level set also being invariant under the action of B_{4}. We now describe these level sets, first noting that one can show that on V_{42}^{(2)} the function c′_{43} only takes on values in the range: [−4, 0].

Proposition 7.8. For 0 < t < 4 the set V_{42}^{(2)} ∩ V (c′_{43} + t) is a union of four singular tori. For t = 0, 4 the set V_{42}^{(2)} ∩ V (c′_{43} + t) is a union of two singular tori.

Proof. Assume first that 0 < t < 4. Then solving for a_{ij}(ϵ), 1 ≤ j < i < 4 (for each of the 4 possible ϵ) and substituting into c′_{43} we obtain a degree 4 equation in a_{43}. Solving this equation gives 16 solutions and so we get 16 discs. One calculates the identifications on the boundaries of these discs and sees that four of these form a torus and that this happens 4 times. The cases t = 0, 4 are similar. □

§8 The rank n − 1 case

In this section we assume that R = ℝ. We have already investigated this case for n = 3 in §4.

Theorem 8.1. For all n ≥ 3 there is a one-parameter family of semialgebraic subsets of ℝ^{\binom{n}{2}} each of which is homeomorphic to a smooth sphere of dimension \binom{n}{2} − 1. Each point x of one of these spheres corresponds to a matrix U_{n}(x) which is positive definite. These spheres are invariant under the action of B_{n}. Moreover the kernel of the action of B_{n} on these spheres is the cyclic centre of B_{n} for a dense set of values of the parameter.

Proof. We will be considering the level sets of the function det(U_{n}). First note that the matrix U_{n}((0, 0, …, 0)) corresponding to the origin of ℝ^{\binom{n}{2}} is twice the identity matrix, which is positive definite. Since positive-definiteness is an open condition all matrices corresponding to points in a sufficiently small neighbourhood of the origin of ℝ^{\binom{n}{2}} will be positive definite.

We will use Morse theory applied to the function det(U_{n}) and will need:

Lemma 8.2. The origin is a singular point of det(U_{n}). The Hessian of the function det(U_{n}) at the origin is non-degenerate and has index \binom{n}{2}.

Proof. One easily sees that det(U_{n}) has constant term 2^{n} and that all other monomials have degree at least 2. Thus the origin is a singular point of det(U_{n}). In fact the only monomials of det(U_{n}) having degree 2 are those of the form −2^{n−2}a_{ij}^{2}. Thus the Hessian of det(U_{n}) at the origin is a diagonal matrix with −2^{n−1}s on the diagonal. It is thus non-degenerate and has index \binom{n}{2}. □

Thus Morse theory [GG, II, §6] says that the origin is an isolated singular point and that near the origin det(U_{n}) looks like (after a change of variables):

2^{n} − ∑_{1≤j<i≤n} a_{ij}^{2},

and so the level sets of det(U_{n}) near the origin are all smooth topological spheres of dimension \binom{n}{2} − 1.

Now a dense set of these spheres contain points whose coordinates are algebraically independent. At any such point the kernel of the action of B_{n} is just the cyclic centre of B_{n} [H1]. This proves Theorem 8.1 and a part of Theorem 2 in §1. □

Remark. We note that the level sets of the invariants c′_{ni} intersect these spheres and give in general smaller sets on which the braid groups act.

The case n = 4 can be considered in more detail: What we proved in §4 about the n = 3 case can be summarised as follows: solving det(U_{3}) = −2(c′_{31} + 1) = 0 for a_{21} gives two diffeomorphisms a_{21}^{±}(a_{31}, a_{32}) with domain [−2, 2]^{2} such that the two closed discs a_{21}^{+}([−2, 2]^{2}), a_{21}^{−}([−2, 2]^{2}) meet along a piece-wise linear circle which is a_{21}^{+}(∂[−2, 2]^{2}) = a_{21}^{−}(∂[−2, 2]^{2}), where ∂ denotes the boundary. The union a_{21}^{+}([−2, 2]^{2}) ∪ a_{21}^{−}([−2, 2]^{2}) is a 2-sphere.

Now consider the case n = 4. Here we use Proposition 5.1 to solve the quadratic equation det(U_{n}) = 0 for a_{21}. We obtain two solutions a_{21}^{±}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43}). The same result shows that the discriminant of this equation is 4U_{4[1]}U_{4[2]}. Now by the n = 3 case the equations U_{4[1]}(a_{32}, a_{42}, a_{43}) = 0, U_{4[2]}(a_{31}, a_{41}, a_{43}) = 0 define two 2-spheres in their respective 3-dimensional spaces. Further we have that

a_{21}^{+}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43}) = a_{21}^{−}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43})

at all points where U_{4[1]}(a_{32}, a_{42}, a_{43})U_{4[2]}(a_{31}, a_{41}, a_{43}) = 0. We wish to restrict the domain of a_{21}^{±}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43}) to the closure E_{4} of the component of [−2, 2]^{5} \(U_{4[1]}U_{4[2]})^{−1}(0) containing 0. We will show that this is a 5-ball and that U_{4[1]}(a_{32}, a_{42}, a_{43})U_{4[2]}(a_{31}, a_{41}, a_{43}) is equal to zero only on the boundary of this 5-ball, which is a 4-sphere. Thus

a_{21}^{+}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43})(E_{4}) ∪ a_{21}^{−}m(a_{31}, a_{32}, a_{41}, a_{42}, a_{43})(E_{4})

will be a union of two 5-balls along a 4-sphere i.e. a 5-sphere.

Now we can also consider U_{4[1]}(a_{32}, a_{42}, a_{43}) as a function U_{4[1]}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43}) and as such U_{4[1]}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43}) = 0 has solution set S^{2} × D^{2}, where the D^{2} corresponds to −2 ≤ a_{31}, a_{41} ≤ 2 and the S^{2} is in the cube −2 ≤ a_{32}, a_{42}, a_{43} ≤ 2. A similar thing happens for U_{4[2]}(a_{31}, a_{32}, a_{41}, a_{42}, a_{43}) = 0. By Theorem 4.1 the S^{2}s in these S^{2} × D^{2}s bound convex 3-balls which contain the origin. These two D^{3} × D^{2}s are now both convex and so intersect in a convex 5-ball with four-sphere boundary as required. This proves a part of Theorem 2.

Theorem 8.3. For all 1 ≤ k ≤ n there is a compact subset of V_{n,k} which is B_{n}-invariant.

Proof. For the moment let us consider solving det(U_{n}) = 0 for the variable a_{21} as given in Proposition 5.1. Then a_{21} is a quadratic function of the remaining coordinates a_{ij} ≠ a_{21}. The discriminant of this quadratic function is 4U_{n[1]}U_{n[2]} (see Proposition 5.1 again) and so a_{21} is not defined at points where 4U_{n[1]}U_{n[2]} < 0.

Consider a ray ρ = ρ(t), t ≥ 0, starting at the origin in the Euclidean space ℝ^{\binom{n}{2}−1} with coordinates (a_{31}, a_{32}, …, a_{n n−1}). Let η_{1}, η_{2} : ℝ^{\binom{n}{2}−1} → ℝ^{\binom{n−1}{2}−1} denote the projections with codomains having coordinates (a_{ij}), i, j ≠ 1 and (a_{ij}), i, j ≠ 2 respectively. At the origin the discriminant 4U_{n[1]}U_{n[2]} is positive, in fact each of U_{n[1]}, U_{n[2]} is positive. Let

ϵ > 0, i = 1, 2. As this ray passes through the cube [−2 − ϵ, 2 + ϵ]^{\binom{n}{2}−1} ⊂ ℝ^{\binom{n}{2}−1} the sign of U_{n[i]}(η_{i}(ρ(t))) will change or at least become 0. This is true for n = 2, 3 and is proved in general by induction on n. Let t_{ρ,i} > 0 denote the first value of t when this happens. Let

D_{n} = \{ρ(t)|t ≤ min(t_{ρ,1}, t_{ρ,2}), t_{ρ,1} ≠ t_{ρ,2}\} ⊂ ℝ^{\binom{n}{2}−1}.

Now the set of rays ρ where t_{ρ,1} = t_{ρ,2} is in the closure of the set of rays with t_{ρ,1} ≠ t_{ρ,2}. Thus the closure \overline{D_{n}} is a compact set of dimension \binom{n}{2} − 1. However \overline{D_{n}} is a subset of the domain of each of the diffeomorphisms a_{21}^{±}(a_{31}, a_{32}, …, a_{n n−1}). The (\binom{n}{2} − 1)-discs which are the images of these two functions meet exactly over points where U_{n[1]}U_{n[2]} = 0, namely over points which are in the image of ∂D_{n}. Thus a_{21}^{+}(D_{n}) ∪ a_{21}^{−}(D_{n}) is the union of two compact sets of dimensions \binom{n}{2} − 1 meeting along a common subset of one dimension less.

Now the action of B_{n} on ℝ^{\binom{n}{2}} fixes the origin. Further each ray ρ ⊂ ℝ^{\binom{n}{2}} starting at the origin must cross a_{21}^{+}(\overline{D_{n}}) ∪ a_{21}^{−}(\overline{D_{n}}), which is the union of the graphs of the functions a_{21}^{+}, a_{21}^{−} with domain \overline{D_{n}}, since the projection of this ray to the ℝ^{\binom{n}{2}−1} with coordinates (a_{31}, a_{32}, …, a_{n n−1}) gives either a point or a ray which eventually passes out of \overline{D_{n}}. The point at which ρ hits a_{21}^{+}(\overline{D_{n}}) ∪ a_{21}^{−}(\overline{D_{n}}) is the first point along ρ at which det(U_{n} + I_{n}) = 0. Thus this compact set of dimension \binom{n}{2} − 1 is clearly invariant under the action of B_{n}. □

§9 Faithfulness of the action of B_{n} on quotients of R′_{n}

Recall that we showed in [H1] that the kernel of B_{n} → Aut(R_{n}) or of B_{n} → Aut(R′_{n}) is the centre Z(B_{n}). The main result of this section is the following:

Theorem 9.1. Let n > 2 and let c ∈ R′_{n}, c ∉ R, be invariant under the action of B_{n}. Assume further that either

(i) c has a_{21}a_{32}a_{43} … a_{n n−1}a_{n1} as a factor of a monomial summand of highest degree; or (ii) c has more than one monomial summand of highest degree.

Then B_{n} acts on the quotient R′_{n}/ < c >, where < c > is the ideal of R′_{n} generated by c. The kernel of this action is Z(B_{n}).

Suppose that I is a B_{n}-invariant ideal in R′_{n} such that for all x ∈ I, x ≠ 0, the number of monomial summands of x of highest degree is greater than 1, then B_{n} acts on the quotient R′_{n}/I and the kernel of this action is Z(B_{n})

Proof. Clearly < c > is invariant under B_{n} and so B_{n} acts on R′_{n}/ < c > as required. Now suppose that α ∈ B_{n}, α ≠ id, is in the kernel of this action. Then there are 1 ≤ j < i ≤ n and k_{ij} ∈ R′_{n}, k_{ij} ≠ 0, such that

α(a_{ij}) = a_{ij} + k_{ij}c.

However Lemma 2.3 shows that α(a_{ij}) has a unique monomial of highest degree and leading coefficient ±1 . This would contradict hypothesis (ii) in Theorem 9.1 which would imply that α(a_{ij}) = a_{ij} + k_{ij}c has more than one monomial of highest degree.

Now assume (i). We will need:

Lemma 9.2. For all 1 ≤ i, j ≤ n the monomial a_{21}a_{32}a_{43} … a_{n n−1}a_{n1} is never a factor of the unique monomial of highest degree in α(a_{ij}).

Proof. Let c_{1}, c_{2}, …, c_{n} be a cut system for the generators x_{1}, x_{2}, …, x_{n}. Thus c_{i} is a vertical arc joining π_{i} to the boundary of D_{n} (above π_{i}) and c_{i} only intersects x_{i} (see Figure 1). Now suppose that a_{21}a_{32}a_{43} … a_{n n−1}a_{n1} is a factor of a monomial of highest degree in α(a_{ij}). Then by Lemma 2.2 we see that each of

(x_{1}^{±1}x_{2}^{±1})^{±1}, (x_{2}^{±1}x_{3}^{±1})^{±1}, …, (x_{n−1}^{±1}x_{n}^{±1})^{±1}, (x_{n}^{±1}x_{1}^{±1})^{±1}

is a subword of the cyclically reduced form of α(x_{i}x_{j}) for some choice of ±1s. We will in fact show that in this case each of

(x_{1}x_{2})^{±1}, (x_{2}x_{3})^{±1}, …, (x_{n−1}x_{n})^{±1}, (x_{n}x_{1})^{±1}

is a subword of the cyclically reduced form of α(x_{i}x_{j}). For suppose that x_{k}x_{k+1}^{−1} is a subword of α(x_{i}x_{j}). Let γ_{ij} denote the simple closed curve containing π_{i} and π_{j} in its interior and representing the conjugacy class x_{i}x_{j}; we assume that α(γ_{ij}) meets the cut-arcs c_{m} minimally. Then we see that α(γ_{ij}) has an oriented subarc δ going from c_{k} on the right to c_{k+1} on the right and such that c_{k} ∪ δ ∪ c_{k+1} cuts off a punctured disc. Now since α(γ_{ij}) is simple and meets the c_{m} minimally we see that the next c_{m} crossed by α(γ_{ij}) is c_{k}. Thus x_{k+1}^{−1}x_{k}^{−1} is a subword of α(x_{i}x_{j}).

The case where x_{k}^{−1}x_{k+1} is a subword of α(x_{i}x_{j}) is similar, as are the cases x_{n}^{−1}x_{1} and x_{n}x_{1}^{−1}.

We thus see that the simple closed curve α(γ_{ij}) has subarcs joining each c_{k} (on the right) to c_{k+1} (on the left) for k = 1, …, n−1 together with subarcs joining c_{n} (on the right) to c_{1} (on the left). Let ζ_{k} be such an arc for each k = 1, …, n and assume further that among all such subarcs the ζ_{k} that we choose is the one closest to the boundary of D_{n}. Then it easily follows that the endpoint of ζ_{k} on c_{k+1} is the end of ζ_{k+1} on c_{k+1}. Thus the ζ_{k}s join up to form a simple closed curve parallel to the boundary. Since n > 2 we see that this cannot be α(γ_{ij}), a contradiction.

The proof of the statement in the last paragraph of Theorem 9.1 is similar to the proof of (ii) in the above. □

Corollary 9.3. The group B_{n}/Z(B_{n}) acts faithfully on the quotient R′_{n}/ < c > in each of the following cases:

(i) c = c_{ni}, for 0 < i < n; (ii) c = det(U_{n}), for n ≥ 3.

Proof. (i) By [H2, Theorem 2.8] we see that the c′_{ni} are invariant under the action of B_{n}. In equation (3.1) a specific matrix is given whose determinant is the characteristic polynomial of T_{1}T_{2} … T_{n} with variable λ. It is clear from the nature of this matrix that for i ≠ 1, n − 1 the coefficient of λ^{i} has more than one monomial of highest degree. The result follows in this case from Theorem 9.1 (ii). If i = 1, n − 1, then we first note that c_{n1} = ±c_{n n−1} and that in this case c_{n1} has a single monomial of highest degree, namely a_{21}a_{32}a_{43} … a_{n n−1}a_{n1}. Theorem 9.1 (i) now concludes this case as well.

(ii) If n > 3, then again c = det(U_{n}) has more than one monomial of highest degree. However if n = 3, then det(U_{3}) = −2(c_{31} + 1) and we are similarly done by Theorem 9.1 (i). □.

This proves Theorem 1 (iii).

We will say that a ring has a Gröbner basis algorithm if the fundamental theorem of Gröbner bases is satisfied for any polynomial ring over that ring. See [AL; Theorem 1.9.1 and Ch. 4] where it is shown that for example ℤ or any field has a Gröbner basis algorithm.

Corollary 9.4. Suppose that R is a ring containing an ideal K such that R/K has characteristic 2 and has a Gröbner basis algorithm. Then for all 1 < r < n the group B_{n}/Z(B_{n}) acts faithfully on the quotient R′_{n}/I_{n r}.

Proof. We first note that we need only consider the case where the coefficients are in R/K since we have natural maps:

B_{n}/Z(B_{n}) → Aut(R′_{n}/I_{nr}) → Aut((R/K)′_{n}/I_{nr})

and if the composition is injective, then so is the first map.

Let α ∈ B_{n} \ Z(B_{n}). If r is even, then I_{nr} = {0} over R/K, but by Lemma 2.3 there are i, j such that α(a_{ij}) has a unique monomial of highest degree greater than one, with coefficient ±1 and so α(a_{ij}) ≠ a_{ij} over R/K.

Now assume that r is odd so that I_{nr} is not trivial. First note that since R/K has characteristic 2 the (r +1)-minors of U_{n} are homogeneous polynomials of degree r +1. Thus the ideals I_{nr} are all homogeneous and have a Gröbner basis algorithm. Thus if ω ∈ I_{nr}, then the homogeneous components of ω are in I_{nr} also. Thus if we have a non-central element in the kernel of the action on (R/K)′_{n}/I_{nr}, then by Lemma 2.3 the ideal I_{nr} must contain a monomial. Thus there cannot be a symmetric matrix over (R/K)′_{n} with 0s on the diagonal and all of whose off-diagonal entries are non-zero, contradicting Proposition 5.2. □

This proves Theorem 1 (iv).

Theorem 9.5. (i) For n ≥ 3 the group B_{n}/Z(B_{n}) does not act faithfully on the quotient R′_{n}/I_{n1}.

(ii) Suppose that R is a ring in which 2 is invertible. Then for n > 3 the group B_{n}/Z(B_{n}) does not act faithfully on the quotient R′_{n}/I_{n2}. In fact if we denote the image of B_{n} in Aut(R′_{n}/I_{n2}) by B_{n2}, then B_{42} fits into a split short exact sequence

0 → ℤ^{2} → B_{42} → B_{3}/ < (σ_{1}σ_{2})^{6} >→ 1,

and so B_{42} has presentation

B_{42} =< σ_{1}, σ_{2}, σ_{3}|σ_{1}σ_{2}σ_{1} = σ_{2}σ_{1}σ_{2}, σ_{1}σ_{3} = σ_{3}σ_{1}, (σ_{1}σ_{2})^{6}, (σ_{1}σ_{3}^{−1}, σ_{2}σ_{1}σ_{3}^{−1}σ_{2}^{−1}) > .

Proof. (i) One checks that σ_{1}^{4} acts trivially on R′_{n}/I_{n1}. Generators for the ideal I_{n1} were given in Proposition 3.3. One needs to check the two cases where the characteristic of R is or is not 2. This is straightforward. This, together with the results of §3, proves Theorem 1 (i).

(ii) It is easy to check that if α is the Dehn twist about the curve surrounding the first three punctures, so that α = (σ_{1}σ_{2})^{3}, then α^{2} is in the kernel of this representation. For example for 3 < r ≤ n we have

α^{2}(a_{r1}) = a_{r1} + a_{r1}U_{n}([1, 2, 3])^{2}/4 − 5a_{r1}U_{n}([1, 2, 3], [2, 3, r])/2
+ (a_{r1}a_{32}^{2} − a_{r1}a_{31}a_{32} + a_{r2}a_{21} + a_{r3}a_{31})U_{n}([1, 2, 3])/2.

Since n > 3 we see that α^{2} ∉ Z(B_{n}). This proves Theorem 1 (ii).

For the n = 4 case we need to prove (a) that the image of the rank 2 free subgroup < β_{1} = σ_{1}σ_{3}^{−1}, β_{2} = σ_{2}σ_{1}σ_{3}^{−1}σ_{2}^{−1} >⊂ B_{4} [Bi] in B_{42} is isomorphic to ℤ^{2}; and (b) that there are no other relations in the image of the subgroup < σ_{1}, σ_{2} > in B_{42} other than the ones we have already listed.

Now [GL] there is a split short exact sequence 1 → F_{2} → B_{4} → B_{3} → 1, where F_{2} is the free group of rank 2 freely generated by β_{1}, β_{2}. Now one also checks that β_{1}β_{2}β_{1}^{−1}β_{2}^{−1} has trivial action on R′_{4}/I_{42} (but not on R′_{n}/I_{n2} for n > 4). To proceed we need to show that the action of β_{1}^{r}β_{2}^{s} on R′_{4}/I_{42} is non-trivial for all r, s ∈ ℤ with (r, s) ≠ (0, 0). It will suffice to find a homomorphism ϕ : R′_{4} → ℝ such that ϕ(U_{4}) has rank 2 and such that ϕ(β_{1}^{r}(U_{4})) ≠ ϕ(β_{2}^{s}(U_{4})) for all such r, s. In fact we will show that the 12 entries of ϕ(β_{1}^{r}(U_{4})) and ϕ(β_{2}^{s}(U_{4})) are different. Let ϕ be determined by:

ϕ(U_{4}) = ( \begin{matrix}2 & 3/2 & 3/2 & \sqrt{14}/2 \\ 3/2 & 2 & 1/4 & \sqrt{14}/4 \\ 3/2 & 1/4 & 2 & \sqrt{14}/2 \\ \sqrt{14}/2 & \sqrt{14}/4 & \sqrt{14}/2 & 2 \end{matrix} ) .

Note also that we may without loss assume that r, s are both even. Now we note that using (1.2) one can show that β_{1}^{2} acts on U_{4} as follows: β_{1}^{2}(U_{4}) = EU_{4}E^{t} where E is the matrix

E = ( \begin{matrix}1 − a_{21}^{2} & a_{21} & 0 & 0 \\ −a_{21} & 1 & 0 & 0 \\ 0 & 0 & 1 & −a_{43} \\ 0 & 0 & a_{43} & 1 − a_{43}^{2} \end{matrix} ) .

It follows that the 12 entry of β_{1}^{2}(U_{4}) = EU_{4}E^{t} is a_{21} and that similarly the 43 entry is a_{43}. Since β_{1} fixes a_{21} and a_{43} we see that the action of any even power of β_{1} is given by: β_{1}^{2r}(U_{4}) = E^{r}U_{4}(E^{r})^{t} for all r ∈ ℤ. Now

β_{2}^{s}(a_{21}) = σ_{2}β_{1}^{s}σ_{2}^{−1}(a_{21}) = σ_{2}β_{1}^{s}(a_{31})

and so if β_{2}^{s}(a_{21}) = a_{21}, then we would have β_{1}^{s}(a_{31}) = a_{31} and so ϕ(β_{1}^{s}(a_{31})) = ϕ(a_{31}). Now

write E = ( \begin{matrix}E_{1} & 0 \\ 0 & E_{2} \end{matrix} ), where E_{1}, E_{2} are 2 × 2 matrices. Then putting U_{4} = ( \begin{matrix}A & B \\ C & D \end{matrix} ) we

have (where r = 2w):

β_{1}^{r}(U_{4}) = E^{w}U_{4}(E^{w})^{t} = ( \begin{matrix}E_{1}^{w}A(E_{1}^{w})^{t} & E_{1}^{w}B(E_{2}^{w})^{t} \\ E_{2}^{w}C(E_{1}^{w})^{t} & E_{2}^{w}D(E_{2}^{w})^{t} \end{matrix} ) .  (9.1)

Thus we need to show that the 11 entry of E_{1}^{w}B(E_{2}^{w})^{t} is never 3/2 for w ≠ 0. We assume that there is a w ≠ 0 such that the 11 entry of E_{1}^{w}B(E_{2}^{w})^{t} is 3/2. Now let us order the

entries of the matrix B = ( \begin{matrix}a_{31} & a_{41} \\ a_{32} & a_{42} \end{matrix} ) as (a_{31}, a_{41}, a_{32}, a_{42}). Then relative to this ordering

the action of E on B given by the 12 block of (9.1) can be described by the following matrix:

H = ( \begin{matrix}−5/4 & 5/8 \sqrt{14} & 3/2 & −3\sqrt{14}/4 \\ −5/8 \sqrt{14} & \frac{25}{8} & 3\sqrt{14}/4 & −\frac{15}{4} \\ −3/2 & 3\sqrt{14}/4 & 1 & −\sqrt{14}/2 \\ −3\sqrt{14}/4 & \frac{15}{4} & \sqrt{14}/2 & −5/2 \end{matrix} )

so that the entries of E_{1}^{w}B(E_{2}^{w})^{t} are given by H^{w}(a_{31}, a_{41}, a_{32}, a_{42})^{t}. Now H has 4 distinct eigenvalues and expressing the vector (3/2, \sqrt{14}/2, 1/4, \sqrt{14}/4) as a linear combination of these eigenvectors we see that the 11 entry of E_{1}^{w}B(E_{2}^{w})^{t} is

\frac{(3 − i)}{4}(−\frac{9}{16} + \frac{5}{16}i\sqrt{7})^{w} + \frac{(3 + i\sqrt{7})}{4}(−\frac{9}{16} − \frac{5}{16}i\sqrt{7})^{w}

= 3/2 cos(w (arctan(5/9 \sqrt{7}) − π)) − \sqrt{7}/2 sin(w (arctan(5/9 \sqrt{7}) − π)), (9.2)

where i^{2} = −1. Now let x = cos(w (arctan(5/9 \sqrt{7}) − π)) and solve for the expression (9.2) equal to 3/2. This gives x = 1, \frac{1}{8}. Now if we have cos(w (arctan(5/9 \sqrt{7}) − π)) = 1, then arctan(5/9 \sqrt{7}) − π = 2nπ/w for some n ∈ ℤ. Thus tan(2nπ/w) = 5\sqrt{7}/9, from which we get that cos(2nπ/w) = ±9/16. Now from [CJ, Theorem 7] we see that there is no rational multiple of π whose cosine is ±9/16, a contradiction.

Now assume that cos(w (arctan(5/9 \sqrt{7}) − π)) = 1/8. Then a similar computation shows that cos(arccos(\frac{1}{8})/w) = ±9/16. Now we may assume that w ≥ 1 and if we think of w as a real variable, then g(w) = cos(arccos(\frac{1}{8})/w) is a real-valued function and as such is strictly increasing on the domain [1, ∞). One checks that on this domain g(w) is positive and that 1 < g^{−1}(9/16) < 2; thus there is no integral w with g(w) = ±9/16. This shows that the subgroup < β_{1}, β_{2} > of B_{4} is represented in Aut(R′_{4}/I_{42}) as ℤ^{2} as required.

The remainder of the proof of Theorem 9.5 follows from the case n = 4 of the following result, requiring the calculation and use of a Gröbner basis, for which we use the computer algebra system MAGMA.

Proposition 9.6. Let n = 4, 5, 6, 7, 8. Then the image of the group B_{3}/ < (σ_{1}σ_{2})^{6} > in B_{n}/ < (σ_{1}σ_{2})^{6} > contains no elements in the kernel of the action on R′_{n}/I_{n2}.

Proof. Let α ∈ B_{3} < B_{n} be in the kernel. Now one can check that the generator β = (σ_{1}σ_{2})^{3} of the cyclic centre of B_{3} has non-trivial action on R′_{n}/I_{n2}, n = 4, 5, 6, 7, 8, using MAGMA [MA] (one calculates the ideal I_{n2}, n = 4, 5, 6, 7, 8 and shows that β(a_{41}) − a_{41} ∉ I_{n2} by computing a Gröbner basis for I_{n2} for each given value of n, and then showing that the normal form of β(a_{41}) − a_{41} is non-zero relative to this Gröbner basis). However, if α ∈ B_{3} \ Z(B_{3}), then α acts non-trivially on R′_{3} i.e. there are 1 ≤ i ≠ j ≤ 3 such that d = α(a_{ij}) − a_{ij} ≠ 0. Since α is in the kernel of the action on R′_{n}/I_{n2}, then we must have d ∈ I_{n2}∩R′_{3}. But I_{n2}∩R′_{3} is an elimination ideal [AL] and a calculation using the elimination ideal routine algorithm in MAGMA shows that I_{n2} ∩ R′_{3} =< c′_{31} > for n = 4, 5, 6, 7, 8. Thus we would have d = kc′_{31} for some k ∈ R′_{3}, k ≠ 0, contradicting Corollary 9.3. The result follows. □

Remark 9.7. Of course we conjecture that I_{n2} ∩ R′_{3} =< c′_{32} > for all n ≥ 4, which would in turn show that Proposition 9.6 was true for all n ≥ 4.

Remark 9.8. We note that B_{3}/ < (σ_{1}σ_{2})^{6} >≅ SL(2, ℤ) and so it should not be surprising to find that one can prove that B_{42} ≅< m_{1}, m_{2}, m_{3} > is a subgroup of SL(3, ℂ) where

m_{1} = ( \begin{matrix}1 & 1 & c \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) , m_{2} = ( \begin{matrix}1 & 0 & 0 \\ −1 & 1 & f \\ 0 & 0 & 1 \end{matrix} ) , m_{3} = ( \begin{matrix}1 & 1 & a \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) .

Here a, c, f ∈ ℂ must satisfy a ≠ c for this linear representation to be faithful.

§10 Action of B_{n}

In this section we will want to think of the exp(tD(α)) as acting as automorphisms of power series rings (the power series rings associated to R_{n} and R′_{n}) and also as acting on the Euclidean space ℝ^{\binom{n}{2}}. We now explain the first of these.

Let R = ℂ[x_{1}, …, x_{n}] be a polynomial algebra and let R^{∗} = ℂ[[x_{1}, …, x_{n}]] be the corresponding ring of formal power series. Let W_{n} be the general Lie algebra of Cartan type, i.e. W_{n} is linearly generated by all ℂ-derivations of the form

f \frac{∂}{∂x_{i}} , i = 1, …, n

where f ∈ R^{∗} [Ca, SS, Ka]. Now W_{n} (and so any of its subalgebras) can be given a filtration

W_{n} = W_{n,−1} ⊃ W_{n,0} ⊃ W_{n,1} ⊃ ⋯ ⊃ W_{n,i} ⊃ …

Here an element of W_{n,i} is a linear sum of f \frac{∂}{∂x_{j}} ’s where f has degree at least i + 1. There is a corresponding graded algebra with graded pieces W_{n,i}/W_{n,i+1}. Using this grading we can define a valuation on W_{n} as follows (c.f. [J, p.171]): |0| = 0, and if a ≠ 0, then |a| = 2^{−i} where a ∈ W_{n,i} and a ∉ W_{n,i+1}. Then we have:

|a| ≥ 0,  |a| = 0 if and only if a = 0,
|[a, b]| ≤ |a||b|, and |a + b| ≤ max{|a|, |b|}.

This makes W_{n} into a topological algebra where a_{1} + a_{2} + … converges if and only if |a_{i}| → 0 as i → ∞.

For any derivation D ∈ W_{n} and any x ∈ R^{∗} we define

exp(D)(x) = x + D(x) + \frac{1}{2!}D^{2}(x) + \frac{1}{3!}D^{3}(x) + ….

In [H3] we proved that each D(α) converges in the above topology for any α ∈ P_{n}. Thus the groups P_{n} and B_{n} are also well-defined.

Next we note that the action of α ∈ P_{n} on the ring R′_{n} is also given as exp(D(α)), where we can just ignore all the \frac{∂}{∂a_{ij}} for i < j and replace a_{ij} by −a_{ji} for i < j in D(α).

We remarked in §1 that each such D(α) has the property that D(α)(c_{ni}) = 0 for all α ∈ P_{n} and all i ≤ n. This easily implies that each exp(tD(α)), α ∈ P_{n} fixes each of the c_{ni}.

In order to study the effect of exp(tD(α)), α ∈ P_{n}, on the real varieties V_{nk} we need to see that the derivation D(α) with the a_{ij} replaced by real numbers converges. Since for |x| < 1 we have

arcsinh(x) =x − \frac{1}{2 · 3}x^{3} + \frac{1 · 3}{2 · 4 · 5}x^{5} − \frac{1 · 3 · 5}{2 · 4 · 6 · 7}x^{7}+

⋯ + (−1)^{n} \frac{1 · 3 · 5 · . . . · (2n − 1)}{2 · 4 · 6 · . . . · (2n) · (2n + 1)}x^{2n+1} + …

(with a similar expression in the case |x| > 1) we see that arcsinh(x)/x converges everywhere in ℝ. Thus the exponential exp(tD(σ_{1}^{2})) given in §1 converges for all values of t ∈ ℝ. It follows similarly that each exp(tD(α)) is well-defined for any α ∈ P_{n} and at any point in ℝ^{\binom{n}{2}}. Thus the groups P_{n} and B_{n} are also well-defined as sets of invertible functions acting on ℝ^{\binom{n}{2}}.

Now each exp(tD(α)), α ∈ P_{n}, t ∈ ℝ, acts as an automorphism of the power series ring R̄′_{n} and so acts on the matrix U_{n} in such a way as to preserve rank. Thus the real algebraic subvarieties V_{nk}, V_{nk}^{(2)} are fixed by the action of P_{n} and so by B_{n}. This proves Theorem 3.

Proposition 10.1. The action of exp(tD(σ_{1})) is given as follows:

a_{21} ↦ a_{21};
a_{r1} ↦ a_{r1}cosh(tα_{21}) − (2a_{r2} − a_{r1}a_{21})sinh(tα_{21})/\sqrt{a_{21}^{2} − 4};
a_{r2} ↦ a_{r2}cosh(tα_{21}) + (2a_{r1} − a_{r2}a_{21})sinh(tα_{21})/\sqrt{a_{21}^{2} − 4}.

Here r > 2 and α_{21} = arcsinh(\sqrt{a_{21}^{4}/4 − a_{21}^{2}}).

Proof. From the expression for D(σ_{1}^{2}) given in §1 we can prove by induction that

D(σ_{1}^{2})^{i}(a_{r1}) = a_{r1}t^{i}a_{21}^{i}α_{21}^{i}(a_{21}^{2} − 4)^{i/2}/2^{i} for i ∈ 2ℕ;
D(σ_{1}^{2})^{i}(a_{r1}) = (2a_{r2} − a_{r1}a_{21})t^{i}a_{21}^{i}α_{21}^{i}(a_{21}^{2} − 4)^{(i−1)/2}/2^{i} for i ∉ 2ℕ.

From this we deduce the image of a_{r1}. There is a similar expression for D(σ_{1}^{2})(a_{r2}) giving the image of a_{r2}. □

For 1 ≤ i < n the action of σ_{i}^{2} can easily be deduced from this.

We recall Thurston’s earthquake theorem, namely that earthquakes act transitively on the Teichmüller T_{g,0,0}; for definitions and results about earthquakes see [Ke1, Ke2, Th]. In our situation we see that an earthquake does not act transitively on T_{0,n,1} since an earthquake can’t change the length of a boundary geodesic.

Conjecture 10.2. The action of exp(D(σ_{1}^{2}) given in 10.1 is the action on Teichmüller space of the earthquake along the curve x_{1}x_{2}. The action of B_{n} is transitive on level sets.

.

§11 Teichmüller space

In this section we have R = ℝ. Basic facts about Teichmüller spaces and Fuchsian groups can be found in [IT]. The punctured disc D_{n} can be represented as the quotient ℍ^{2}/G where G is the Fuchsian group generated by elements s_{1}, s_{2}, …, s_{n} where s_{2}, …, s_{n} are hyperbolic and p_{1} = s_{1}^{−1}, p_{2} = s_{2}^{−1}s_{1}, p_{3} = s_{3}^{−1}s_{2}, …, p_{n} = s_{n}^{−1}s_{n−1} are all parabolic. See Figure 5 to see how the identifications occur.

In this diagram (for the case n = 4) the parabolic matrices p_{i}, i ≤ n, have fixed points f_{i} as shown. The hyperbolic element s_{i}, i > 1, takes the geodesic joining f_{i} to f_{i+1} to a geodesic joining g_{i} to g_{i+1} as indicated (here g_{1} = f_{1}). Now up to conjugation in P SL_{2}(ℝ) we may assume that

p_{1} = s_{1}^{−1} = ( \begin{matrix}−1 & 0 \\ u & −1 \end{matrix} ) , p_{2} = s_{2}^{−1}s_{1} = ( \begin{matrix}−1 − v & v \\ −v & v − 1 \end{matrix} ) ,

p_{3} = s_{3}^{−1}s_{2} = ( \begin{matrix}−1 & w \\ 0 & −1 \end{matrix} ) ,  (11.1)

so that p_{1}(0) = s_{1}(0) = 0 = f_{1}, p_{2}(1) = 1 = f_{2} and p_{3}(∞) = ∞ = f_{3}. These three conditions are the only control that we have on the parabolics listed above (since an element of P SL_{2}(ℝ) is completely determined by its action on any ordered triple of elements of ℝ ∪ {∞}) and so determine a ‘normalised’ set of generators. The rest of the parabolics must then have the form:

p_{i} = s_{i}^{−1}s_{i−1} = ( \begin{matrix}x_{i1} & x_{i2} \\ −(1 + x_{i1})^{2}/x_{i2} & −2 − x_{i1} \end{matrix} ) ,  (11.2)

for 3 < i ≤ n; here x_{i2} ≠ 0 since p_{i} does not fix 0. This gives 2n − 3 unknowns u, v, w, x_{ij}. Note that the real numbers f_{i}, g_{i} must satisfy:

f_{4} < f_{5} < ⋯ < f_{n} < f_{n+1} < g_{n+1} < g_{n} < ⋯ < g_{3} < g_{2} < g_{1} = f_{1} = 0.

In fact given any n parabolics as in (11.1) and (11.2) with the first three fixing 0, 1, ∞, then we obtain a Fuchsian group G such that ℍ^{2}/G ≅ D_{n} if the above inequalities are satisfied. This thus gives a set of coordinates u, v, w, x_{41}, x_{42}, …, x_{n1}, x_{n2} for the Teichmüller space T_{0,n,1}.

Example 11.1. In the case n = 3 we need f_{4} < g_{4} < g_{3} < g_{2} < g_{1} = f_{1} = 0. But one can verify that

g_{3} = \frac{v − 1}{uv + v − u}, g_{4} = \frac{−2 v − 2 uwv − wv + uv + uw + uwv^{2} + wv^{2} − uv^{2}}{(−u + uv + v) (uwv + wv − uv − uw − 2)} ,

g_{2} = \frac{1}{u + 1}; f_{4} = −1/2 \frac{−2 v + uwv + wv − uv − uw}{−u + uv + v} .

Thus we get the inequalities

u < −1, uv + v − u < 0, uvw + vw − uv − uw < 0.  (11.3)

Of course the matrices p_{i}, s_{i} are only defined up to a sign, however Keen [K1 pp. 210-211, K2] notes that when choosing representative matrices from SL_{2}(ℝ) for the elements of P SL_{2}(ℝ), there is a canonical choice for the signs of the traces of the generators, namely we may take them all to be negative, so that in our case we choose trace(p_{i}) = −2. We will also need to note that s_{i}^{−1} = p_{i}p_{i−1} … p_{2}p_{1} for i > 1 and that these elements are hyperbolic i.e. they have squared trace greater than 4.

Now it follows from [O, Theorem 4.1] that 2n − 3 of the traces trace(p_{i}p_{j}) completely determine the point of Teichmüller space corresponding to this representation. The 2n − 3 traces given in [O] are those of p_{1}p_{i}, i = 2, …, n and p_{2}p_{i}, i = 3, …, n.

As indicated in equation (1.4), we would like to solve trace(p_{i}p_{j}) = 2 − a_{ij}^{2}. Now we see that there are 2n − 3 of the u, v, w, x_{ij}, but \binom{n}{2} of the a_{ij} and so in general there is no hope of solving for the u, v, w, x_{ij} as functions of the a_{ij}, except when n = 3. Also, since s_{2}, …, s_{n} and the p_{i}p_{j}, i ≠ j are all hyperbolic we are interested in having

|trace(p_{1}p_{2})|, |trace(p_{1}p_{3})|, |trace(p_{2}p_{3})|, …, |trace(p_{1}p_{2}p_{3})|, ⋯ > 2.

Consider the case n = 3 again. Solving trace(p_{i}p_{j}) = 2 − a_{ij}^{2} for u, v, w we get two solutions:

\{w = \frac{a_{31} a_{32}}{a_{21}} , v = \frac{a_{32} a_{21}}{a_{31}} , u = −\frac{a_{31} a_{21}}{a_{32}} \} , \{w = −\frac{a_{31} a_{32}}{a_{21}} , v = −\frac{a_{32} a_{21}}{a_{31}} , u = \frac{a_{31} a_{21}}{a_{32}} \} .

Substituting into the above equalities we get:

∓ \frac{a_{31} a_{21} ∓ a_{32}}{a_{32}} < 0; ±\begin{matrix}a_{21} & (a_{31}^{2} ∓ a_{21} a_{31} a_{32} + a_{32}^{2}) \\ a_{31} & a_{32} \end{matrix} < 0;

∓ a_{21} a_{31} a_{32} + a_{32}^{2} + a_{21}^{2} + a_{31}^{2} < 0.

Here we consistently take the top or bottom choice of signs in these equations, corresponding to the two solutions given in the previous paragraph. We note that this shows that points of T̄_{0,3,1} lie in V_{t} with t < 0.

From the above we see that (when n = 3) there are points of Teichmüller space with |trace(p_{1}p_{2})| > 2; thus with |a_{21}|, |a_{31}|, |a_{32}| > 2 (in fact they can be arbitrarily large). Thus in each component of T̄_{0,3,1} ⊂ ℝ^{3} that covers T_{0,3,1} we must have points (a_{21}, a_{31}, a_{32}) satisfying:

|a_{21}|, |a_{31}|, |a_{32}| > 2 and trace(p_{3}p_{2}p_{1}) = a_{21}^{2} + a_{31}^{2} + a_{32}^{2} − a_{21}a_{31}a_{32} − 2 ≠ 2.

Now let P be the component of the complement of the surface defined by a_{21}^{2} + a_{31}^{2} + a_{32}^{2} − a_{21}a_{31}a_{32} −4 = 0 which contains the point (3, 3, 3). Note that P is an open 3-ball. Then one can also check that P contains a part of T̄_{0,3,1}. Not every point of P will represent a point of Teichmüller space; in fact only those satisfying the conditions corresponding to (11.3) will do so. We will use these facts later.

In order to prove Theorem 6 and the first part of Theorem 4 we need to show that the actions of B_{n} on both sides of the equation 2 − trace(p_{i}p_{j}) = a_{ij}^{2}, i > j, are compatible at least for some set of lifts of the a_{ij} which are permuted by the B_{n} action. More precisely, for α ∈ B_{n} suppose that α(a_{ij}) = f_{α,i,j}(a_{21}, a_{31}, …, a_{nn−1}). Then what we need to do amounts to finding all ϵ_{ij} ∈ {±1}, i > j, such that for all β ∈ B_{n} there are δ_{ij} ∈ {±1}, n ≥ i > j ≥ 1, so that if a_{ij} = ϵ_{ij}\sqrt{2 − trace(p_{i}p_{j})} then

f_{β,i,j}(ϵ_{21}a_{21}, ϵ_{31}a_{31}, …, ϵ_{nn−1}a_{nn−1}) = δ_{ij}\sqrt{2 − trace(β(p_{i})β(p_{j}))}.  (11.5)

To proceed we will need to note the following trace identity for three non-pair-wise commuting parabolics p_{1}, p_{2}, p_{3} with trace −2:

\sqrt{2 − trace(p_{1}p_{2}p_{1}^{−1}p_{3})} = \sqrt{(2 − trace(p_{1}p_{2}))(2 − trace(p_{1}p_{3}))}−\sqrt{2 − trace(p_{2}p_{3})}. (11.6)

This is easily checked since we can conjugate p_{1}, p_{2}, p_{3} so that they are as in (11.1) and then easily check this identity (recall that parabolics commute if and only if they have the same fixed point).

We reduce immediately to the case where β ∈ B_{n} is a generator: β = σ_{i}; and the action of β is given by (1.2) only where we have a_{ij} = −a_{ji}. Now β(a_{i+1i}) = −a_{i+1i} and so (11.5) gives

−ϵ_{i+1i}\sqrt{2 − trace(p_{i+1}p_{i})} = δ_{i+1i}\sqrt{2 − trace(p_{i+1}p_{i})},

so that we have −ϵ_{i+1i} = δ_{i+1i} for all 1 ≤ i < n.

Next for j < i we have β(a_{i+1j}) = a_{ij} and so (11.5) gives ϵ_{ij}\sqrt{2 − trace(p_{i}p_{j})} = δ_{i+1j}\sqrt{2 − trace(p_{i}p_{j})}, which gives ϵ_{ij} = δ_{i+1j}.

Keeping j < i we also have β(a_{ij}) = a_{i+1j} − a_{i+1i}a_{ij}, so that (11.5) and (11.6) give

ϵ_{i+1j}\sqrt{2 − trace(p_{i+1}p_{j})} − ϵ_{i+1i}ϵ_{ij}\sqrt{(2 − trace(p_{i+1}p_{i}))(2 − trace(p_{i}p_{j}))}
= δ_{ij}\sqrt{2 − trace(p_{i}p_{i+1}p_{i}^{−1}p_{j})}
= δ_{ij} (\sqrt{(2 − trace(p_{i+1}p_{i}))(2 − trace(p_{i}p_{j}))} − \sqrt{2 − trace(p_{i+1}p_{j})}) ,

showing that ϵ_{i+1j} = −δ_{ij} and ϵ_{i+1i}ϵ_{ij} = −δ_{ij}.

Continuing this analysis for β(a_{ji}) = a_{ji+1} − a_{ji}a_{i+1i} and β(a_{ji+1}) − a_{ji} we obtain the following equations relating the ϵ_{ij} and δ_{ij}:

(i) ϵ_{i+1i} = −δ_{i+1i} for all 1 ≤ i < n;
(ii) ϵ_{ij} = δ_{i+1j} for all 1 ≤ j < i ≤ n;
(iii) ϵ_{i+1j} = −δ_{ij} for all 1 ≤ j < i ≤ n;
(iv) ϵ_{i+1i}ϵ_{ij} = −δ_{ij} for all 1 ≤ j < i ≤ n;
(v) ϵ_{ji} = δ_{ji+1} for all 1 < i + 1 < j ≤ n;
(vi) ϵ_{ji+1} = −δ_{ji} for all 1 < i + 1 < j ≤ n;
(vii) ϵ_{ji}ϵ_{i+1i} = −δ_{ji} for all 1 < i + 1 < j ≤ n.

From (iii) and (iv) and from (vi) and (vii) we get

ϵ_{ji+1}ϵ_{ji}ϵ_{i+1i} = 1, ϵ_{i+1j}ϵ_{ij}ϵ_{i+1i} = 1.

In fact from the above relations it is apparent that these are the only relations (for fixed i) satisfied by the ϵ_{rs}. Further, from ϵ_{i+1j}ϵ_{ij}ϵ_{i+1i} = 1 and (i), (ii) and (iii) we see that we must have δ_{i+1j}δ_{ij}δ_{i+1i} = 1, and that similarly we must have δ_{ji+1}δ_{ji}δ_{i+1i} = 1 following from (i), (v) and (vi).

Since we are interested in having these relations for all i < n one checks that for any ϵ_{rs} satisfying ϵ_{ij}ϵ_{jk}ϵ_{ik} = 1 for all i > j > k there are δ_{rs} satisfying δ_{ij}δ_{jk}δ_{ik} = 1 for all i > j > k which solve (i)-(vii). But any such ϵ_{rs} are clearly determined by n − 1 of them, namely ϵ_{n1}, ϵ_{n2}, …, ϵ_{nn−1} and that we must have

ϵ_{rs} = ϵ_{nr}ϵ_{ns} for all n > r > s ≥ 1.

Similar conditions hold for the δ_{rs}. Thus we see that we get T̄_{0,n,1}, a cover of T_{0,n,1} (in the broad sense) consisting of points (a_{ij}) over T_{0,n,1} where sign(a_{ij}a_{jk}a_{ik}) = 1 for all i > j > k. This gives 2^{n−1} disjoint copies of T_{0,n,1} embedded smoothly in ℝ^{\binom{n}{2}} with coordinate functions a_{21}, a_{31}, …, a_{nn−1} such that the action of B_{n} on these 2^{n−1} copies of T_{0,n,1} is by polynomial automorphisms, as in (1.2). This completes the proof of Theorem 6 and part of Theorem 4. □

We note that using the natural coordinates u, v, w, x_{41}, x_{42} for T_{0,4,1} we have the following action of B_{4} as rational automorphisms (the action of B_{3} on the natural coordinates for T_{0,3,1} is obtained by restricting σ_{1} and σ_{2} to u, v, w):

σ_{1}(u, v, w, x_{41}, x_{42}) = (v(1 − u), \frac{u}{1 − u}, w(u − 1),

\frac{x_{41}x_{42}(u − 1) − (1 + x_{41})^{2}}{x_{42}(u − 1)} , \frac{(1 + x_{41} + x_{42} − ux_{42})^{2}}{x_{42}(u − 1)} ),

σ_{2}(u, v, w, x_{41}, x_{42}) = (−u(1 + v), −w(1 + v), \frac{−v}{1 + v} , x_{41} + x_{42}, −x_{42}/(1 + v)),

σ_{3}(u, v, w, x_{41}, x_{42}) = (\frac{u(w + wx_{41} + x_{42})}{1 + w + wx_{41} + x_{41} + x_{42}} , \frac{v(1 + w + wx_{41} + x_{41} + x_{42})}{w + wx_{41} + x_{42}},

\frac{(w + wx_{41} + x_{41} + x_{42} + 1)(w + wx_{41} + x_{42})}{x_{42}}, \frac{−(2wx_{41} + 2w + x_{42})}{w + wx_{41} + x_{42}},

\frac{w(1 + w + wx_{41} + x_{42} + x_{41})}{w + wx_{41} + x_{42}})  (11.4)

This is a non-polynomial action, whereas the action of B_{n} on the a_{ij}-coordinates for T̄_{0,n,1} gives a polynomial action, as we have just indicated.

In what follows we will identify points on all of the 2^{n−1} components of T̄_{0,n,1} by making

all the signs positive and so identify T_{0,n,1} with the positive component T̄_{0,n,1} ∩ ℝ\binom{n}{2}_{>2} of T̄_{0,n,1}.

For the case n = 4 we have 6 of the a_{ij} whereas T_{0,4,1} and T̄_{0,4,1} have dimension 5. Thus the a_{ij} must satisfy some relation on the components of T̄_{0,4,1}. This relation will depend on the choice of signs of the generators p_{i}. We indicate one such relation for the canonical choice of signs in:

Proposition 11.2. Let n = 4 and suppose that p_{1}, p_{2}, p_{3}, p_{4} are all as given in (11.1) and (11.2) with variables u, v, w, x_{41}, x_{42}. Then in the ideal generated by the trace(p_{i}p_{j})−(2−a_{ij}^{2}) there is a single relation satisfied by the a_{ij}:

(a_{21}a_{43} − a_{31}a_{42} − a_{32}a_{41})(a_{21}a_{43} − a_{31}a_{42} + a_{32}a_{41})
×(a_{21}a_{43} + a_{31}a_{42} − a_{32}a_{41})(a_{21}a_{43} + a_{31}a_{42} + a_{32}a_{41}).

The factor a_{21}a_{43} −a_{31}a_{42} +a_{32}a_{41} ∈ R′_{4} is invariant under the action of kerB_{3} → ℤ_{2}, σ_{i} ↦ 1.

Proof. One first calculates the elements trace(p_{i}p_{j}) − (2 − a_{ij}^{2}) and considers the ideal of ℚ[a_{21}, a_{31}, a_{41}, a_{32}, a_{42}, a_{43}, u, v, w, x_{41}, x_{42}] generated by them. One then uses the elimination algorithm [AL, Theorem 2.3.4], as implemented in MAGMA [MA], to eliminate the variables u, v, w, x_{41}, x_{42} and so get the above relation as the only relation satisfied by the a_{ij} (actually, one gets the above relation multiplied by a_{41}^{2}, but this extra factor can be ignored in our case, since we are considering real coefficients). It is easy to check that the generators σ_{i}^{±1}σ_{j}^{±1} of kerB_{3} → ℤ_{2} fix the factor a_{21}a_{43} − a_{31}a_{42} + a_{32}a_{41}. □

The fact that the relation in the above result factors nicely is another indication that Keen’s choice of signs for the generating parabolics really is natural (one can check that other choices of sign do not give relations which so factor).

Of course one can do the above calculation for any n > 3 and obtain an elimination ideal that gives relations that the a_{ij} have to satisfy on the components of T̄_{0,n,1}. For n = 5 one obtains an ideal with 15 generators, five of which look like the one given in Proposition 11.2.

Note that the equation x^{2} +y^{2} +z^{2} −xyz = t is related to the Markoff equation x^{2} +y^{2} + z^{2} = 3xyz, since if one substitutes 3x, 3y, 3z for x, y, z, then it gives x^{2}+y^{2}+z^{2}−3xyz = t/9. The general theory of the Markoff equation [CF] tells us that all positive integral solutions are related to each other via the Markoff tree and an action of a group on the set of solutions.

Theorem 11.3. Let t > 4 and let V_{t} denote the surface a_{21}^{2} + a_{31}^{2} + a_{32}^{2} − a_{21}a_{31}a_{32} − t = 0 in ℝ^{3}. Then B_{3}\V_{t} is compact. Further on the level sets V_{t}, t > 4, there are points with infinite stabilisers.

Proof. First note that we have already described V_{t} for t = 4 (in §4) and have indicated that for t < 4, V_{t} has 4 unbounded components, each a topological disc, while for t > 4, V_{t} is homeomorphic to a sphere with 4 holes. Let x = a_{21}, y = a_{31}, z = a_{32}. The first statement in this result will follow from the next result which is based on ideas of [Mo, pp. 106-110]; in fact the next result is, for our equation, a stronger version of [Mo; Theorem 8 p. 107].

Lemma 11.4. For t > 4 and (x, y, z) ∈ V_{t} ⊂ ℝ^{3} there is α ∈ B_{3} such that α(x, y, z) = (u, v, w) where u^{2} + v^{2} + w^{2} ≤ t.

Proof. One first checks that the elements μ_{1}, μ_{2} where μ_{1}(x, y, z) = (−x, −y, z), μ_{2}(x, y, z) = (x, −y, −z) generate a subgroup M_{3} of Diff(ℝ^{3}), the group of diffeomorphisms of ℝ^{3}, having order 4. We note that M_{3} fixes each V_{t}. Regarding B_{3} as giving a subgroup of Diff(ℝ^{3}) one sees easily that B_{3} normalises M_{3}. Thus we may without loss assume in what follows that any (x, y, z) ∈ V_{t} has x, y ≥ 0.

Note that if we have a solution (x, y, z) with xyz = 0, then x^{2} + y^{2} + z^{2} = t in this case and so we are done. Thus in what follows we may assume xyz ≠ 0, so that with the above we may assume x, y > 0.

Now we have

σ_{1}(x, y, z) = (−x, z − xy, y), σ_{2}(x, y, z) = (y − xz, x, −z)
σ_{1}^{−1}(x, y, z) = (−x, z, y − xz), σ_{2}^{−1}(x, y, z) = (y, x − yz, −z).

We let ||(x, y, z)|| = x^{2} + y^{2} + z^{2}. Now given p = (x, y, z) ∈ ℝ^{3} we replace p by σ_{k}^{ϵ}(p) (for any choice of k = 1, 2, ϵ = ±1) if ||σ_{k}^{ϵ}(p)|| < ||p|| and continue doing this until we have ||σ_{k}^{ϵ}(p)|| ≥ ||p|| for all k = 1, 2, ϵ = ±1. Clearly this is a finite process. We claim that for such a p we must have ||p|| ≤ t. Suppose not for some t and p = (x, y, z). Then x^{2} + y^{2} + z^{2} − xyz = t and ||p|| = x^{2} + y^{2} + z^{2} > t yields xyz > 0, which, together with x, y > 0 gives z > 0.

Now ||σ_{k}^{ϵ}(p)|| ≥ ||p||, for all k = 1, 2, ϵ = ±1, yields the conditions

x^{2}y^{2} − 2xyz ≥ 0, x^{2}z^{2} − 2xyz ≥ 0, y^{2}z^{2} − 2xyz ≥ 0.

Since x, y, z > 0 this gives

xy − 2z ≥ 0, xz − 2y ≥ 0, yz − 2x ≥ 0.

Now multiplying xy ≥ 2z and yz ≥ 2x gives y^{2} ≥ 4 and we similarly see that x^{2}, y^{2} ≥ 4. Next we note that the equations xy − 2z ≥ 0, xz − 2y ≥ 0, yz − 2x ≥ 0 determine an open convex region R ⊂ ℝ_{≥2}^{3}. This is indicated in Figure 6, where we have drawn the part of the surfaces xy − 2z = 0, xz − 2y = 0, yz − 2x = 0 where x, y, z ≥ 2. The z coordinate is in the vertical direction and so the top component is the one determined by z = xy/2. The region R is that ‘inside’ this cone.

Figure 6.

Figure 7.

In Figure 7 we have added to Figure 6 a part of the surface x^{2} + y^{2} + z^{2} − xyz = t (for t = 5) to indicate how this relates to the region R. We will show that for t > 4 the surface V_{t} does not meets R, a contradiction.

Now the surfaces xz − 2y = 0, yz − 2x = 0 meet along the line where x = ±y, z^{2} = 4, while the surfaces xy − 2z = 0, xz − 2y = 0 meet where x^{2} = 4 and the surfaces xy − 2z = 0, yz − 2x = 0 meet where y^{2} = 4. The surface V_{t} is determined by

z_{±} = (xy ± \sqrt{(x^{2} − 4)(y^{2} − 4) + 4(t − 4)})/2.

Now we need to check for example that if x, y ≥ 2, then z_{+}(x, y) ≥ xy/2, but this is clear. Another case is that when x > y, then we would like to show that z_{−}(x, y) ≤ 2x/y. To see this note that solving the equation 2x/y − z_{−}(x, y) = 0 for y gives

y_{±,±}(x, t) = (±\sqrt{x^{2} + t ± \sqrt{(x^{2} + t)^{2} − 16 x^{2}}}) /\sqrt{2},

where all 4 values of the square roots are allowed. Now y_{++} is above and asymptotic to the line y = x. Thus since x > y in this case we see that 2x/y − z_{−}(x, y) ≥ 0 as required. The y_{−±} cases aren’t allowed, since then y < 0

For the case y_{+−} we see that solving y_{+−}(x, t) = 2 gives t = 4, a contradiction. Thus y_{+−}(x, t) ≠ 2. It follows that y_{+−}(x, t) < 2 for all x ≥ 2, t > 4 and so this case does not occur either. This shows that z_{−}(x, y) ≤ 2x/y when x > y. Since everything is symmetric in the variables x, y, the last case (z_{−}(x, y) ≤ 2y/x when y > x) follows by interchanging x, y in the above. This proves Lemma 11.4 and so the first part of Theorem 11.3.

We now show that there are points of the level sets V_{t}, t > 4, with infinite stabilisers. For example the element σ_{1}^{2} ∈ B_{3} fixes all (a_{21}, a_{31}, a_{32}) = (0, \sqrt{t}cos(θ), \sqrt{t}sin(θ)) ∈ V_{t}. □

Let G_{3} denote the group of homeomorphisms of ℝ^{3} generated by μ_{1}, μ_{2}, σ_{1}, σ_{2}. Theorem 11.3 and Lemma 11.4 imply the following related facts:

Corollary 11.5. Suppose that x^{2} + y^{2} + z^{2} − xyz = t > 4 has an integral solution. Then any integral solution of x^{2} + y^{2} + z^{2} − xyz = t is in the G_{3}-orbit of some (x, y, z), where x, y, z ∈ ℤ and |x|, |y|, |z| ≤ \sqrt{t}.

There are no integral solutions of x^{2} + y^{2} + z^{2} − xyz = t for any t ∈ ℤ, with t ≡ 3 mod 4 or t ≡ 3, 6 mod 9.

For t = 5 every integral solution of x^{2} + y^{2} + z^{2} − xyz = t is in the G_{3}-orbit of (0, 1, 2). For t = 8 every integral solution of x^{2} + y^{2} + z^{2} − xyz = t is in the G_{3}-orbit of (1, 1, −2)

or (0, 2, 2). These orbits are distinct.

For t = 9 every integral solution of x^{2} + y^{2} + z^{2} − xyz = t is in the G_{3}-orbit of (3, 0, 0). The orbit in this case is finite.

Proof. The first statement follows directly from Lemma 11.4. Thus for a given integral t > 4 it is easy to check whether there are any integral solutions. Now if x, y, z ∈ ℤ, then one easily checks that x^{2} + y^{2} + z^{2} − xyz mod 4 ∈ {0, 1, 2} and that x^{2} + y^{2} + z^{2} − xyz mod 9 ∈ {0, 1, 2, 4, 5, 7, 8}

When t = 5 a computer calculation shows that all integral solutions are in the orbit of (0, 1, 2). The proof for t = 8, 9 is similar. We also need to note that the action of G_{3} does not change the gcd of the entries; thus when t = 8 the orbits of (1, 1, −2) and (0, 2, 2) are distinct. □

Remark 11.6. The integral values of t > 4 for which x^{2} + y^{2} + z^{2} − xyz = t has no integral solutions given in the above result do not exhaust such values of t. For example this set also includes

{46, 56, 86, 124, 126, 142, 161, 198, 206, 216, 217}.

This is easily proved using Lemma 11.4.

Lemma 11.7. Any point of V_{t} is in the G_{3}-orbit of some (x, y, z) ∈ ℝ^{3} where

|x − yz| ≥ |x|, |y − xz| ≥ |y|, and |z − xy| ≥ |z|.

Proof. This follows from the argument in the proof of Theorem 11.3. □

Lemma 11.7 will help us determine a fundamental domain for the action of G_{3} on V_{t}, t < 0. As usual we may assume that x, y > 0. Now if z ≤ 0, then 0 > t = x^{2} + y^{2} + z^{2} − xyz > 0.

Thus we may restrict attention to the octant x, y, z > 2. Now if we have yz ≤ x, then we would have

t = x^{2} + y^{2} + z^{2} − xyz ≥ x^{2} + y^{2} + z^{2} − x^{2} = y^{2} + z^{2} > 4,

a contradiction. Thus we must have yz > x and similarly xy > z and xz > y. Putting these together with Lemma 11.7 we see that any point of V_{t}, t < 0, is in the G_{3}-orbit of some (x, y, z) where

xy > 2z, yz > 2x and xz > 2y.

Now the equations xy = 2z, yz = 2x, xz = 2y determine real algebraic varieties which cut out a region of V_{t}. An example of this is shown in Figure 8, which, for t = −7, is the projection onto the xy-plane of these three curves in V_{t}. Here we have used each of the above three equations to determine the variable z and substituted this value of z into x^{2} +y^{2} +z^{2} −xyz −t = 0. This gives an equation which we have solved for y as a function of x and t. One gets the following values for y (since we are only interested in having x, y > 2):

2\sqrt{\frac{x^{2} − t}{x^{2} − 4}}, x\sqrt{\frac{x^{2} − t}{x^{2} − 4}}, \frac{4x}{\sqrt{2 x^{2} + 2 t + 2 \sqrt{x^{4} + 2 x^{2}t + t^{2} − 16 x^{2}}}} ,

\frac{4x}{\sqrt{2 x^{2} + 2 t − 2 \sqrt{x^{4} + 2 x^{2}t + t^{2} − 16 x^{2}}}} .

This gives curves which in this projection are asymptotic to the x = 2, y = 2 and x = y lines. The last two equations correspond to solutions of the case yz = 2x, while the first corresponds to xy = 2z and the second to xz = 2y. More generally, the three ‘spokes’ shown in Figure 8 are each asymptotic (in projection) to one of the lines (i) z = 2, x = y; (ii) y = 2, x = z; (iii) x = 2, y − z.

Figure 8

In Figure 8 the component which is above the y = x line corresponds to the equation xz = 2y (we will denote it by γ_{y}), while the component which is below the y = x line corresponds to the equation 2x = yz (we will denote it by γ_{x}). The component which is asymptotic to the x = 2, y = 2 lines corresponds to the equation 2z = xy (we will denote it by γ_{z}). Note that the latter curve is symmetric relative to the line y = x, while the other two are interchanged by reflection in this line.

Let F_{t} denote the closed region of V_{t} determined in this way. Let α = σ_{1}σ_{2}σ_{1}, β = σ_{1}σ_{2}.

Theorem 11.8. The region F_{t} is a fundamental domain for the action of the subgroup J_{3} =< α, βαβ^{−1}, β^{−1}αβ, μ_{1}, μ_{2} > of index 3 in G_{3} on V_{t}, t < 0. The subgroup < μ_{1}, μ_{2} > is normal in J_{3} and in G_{3} with G_{3}/ < μ_{1}, μ_{2} >≅ B_{3}/Z(B_{3}). The region F_{t} contains three copies of a fundamental region for the action of G_{3}.

Proof. First we note that α and β generate B_{3}, that J_{3} has index 3 in G_{3} and that α^{2} = 1, β^{3} = 1. The normality of < μ_{1}, μ_{2} > has already been noted. Further, using the Reidemeister-Schreier method [MKS] as implemented in MAGMA [MA] one can show that J_{3}/ < μ_{1}, μ_{2} >≅ ℤ_{2} ∗ ℤ_{2} ∗ ℤ_{2}. The second statement is clear and the last follows from the above and the first statement, which we now prove.

Next we see that

β(x, y, z) = (z, −x, −y), α(x, y, z) = (−z, −y + xz, −x).

For (x, y, z), (u, v, w) ∈ ℝ^{3} we define (x, y, z) ∼ (u, v, w) if (x, y, z) and (u, v, w) are in the same orbit under the action of J_{3}. Now, if (x, y, z) ∈ γ_{y}, so that xz = 2y, then

(x, y, z) ∼ (−z, xz − y, −x) = (−z, y, −x) ∼ (y, x, z),

showing that points of the curves in Figure 8 which are asymptotic to the line y = x are identified under the J_{3} action in the same way as a reflection across this line would identify them. Since everything is symmetric in x, y, z we similarly obtain two other such identifications.

Standard arguments for the action of Schottky groups (see for example [Ly; p. 197]) now show that F_{t} is a fundamental domain for the action of J_{3} on V_{t}. □

One can check that the three curves γ_{x}, γ_{y}, γ_{z} are permuted by the action of β: β(γ_{y}) = γ_{z}, β(γ_{z}) = γ_{x}, β(γ_{x}) = γ_{y}. Further α(γ_{y}) = γ_{y} ( since σ_{1}σ_{2}σ_{1}(2a_{31} − a_{21}a_{32}) = −(2a_{31} − a_{21}a_{32})) and α(F_{t}) ∩ F_{t} = γ_{y}.

We can also check that β has exactly one fixed point in F_{t}, namely the point (x_{0}, x_{0}, x_{0}) where x_{0} is the real solution of 3x^{2} − x^{3} − t = 0, namely

x_{0} = \frac{(8 − 4t + 4\sqrt{t(t − 4)})^{2/3} + 4 + 2(8 − 4t + 4\sqrt{t(t − 4)})^{1/3}}{2(8 − 4t + 4\sqrt{t(t − 4)})^{1/3}} .

Since β(x, y, z) ∼ (z, x, y) we see that a fundamental domain for < σ_{1}, σ_{2}, μ_{1}, μ_{2} > is obtained by dividing F_{t} into three pieces, all meeting at the point (x_{0}, x_{0}, x_{0}). Further these three pieces are permuted by β and can be chosen so that they contain part of the curves where V_{t} meets the planes x = y, y = z, z = x. This proves Theorem 5.

We now note two consequences. We have already seen that the curves γ_{x}, γ_{y}, γ_{z} are asymptotic to two of the lines {x = 2, y = z}, {y = 2, x = z}, {z = 2, y = x}. Further, one easily sees from the equations for these curves that all of these curves are contained in the positive octant cut out by the planes x = 2, y = 2, z = 2. Thus if we have an integer point of V_{t}, then it can’t be very far up one of the ‘spokes’ of F_{t} Thus we have:

Corollary 11.9. For t < 0 there are only finitely many B_{3}-orbits of integer solutions to the equation x^{2} + y^{2} + z^{2} − xyz = t. In particular, on each level set V_{t} ∩ (T̄_{0,3,1} ∩ ℝ_{>2}^{3}) of Teichmüller space there are only finitely many B_{3}-orbits of integer points. □

Now B_{3}/Z(B_{3}) ≅ P SL_{2}(ℤ) =< a, b|a^{2}, b^{3} > and the action of B_{3}/Z(B_{3}) on the quotient < μ_{1}, μ_{2} > \V_{t} looks exactly like the action of the index 3 subgroup J′_{3} =< a, bab^{−1}, b^{−1}ab > of P SL_{2}(ℤ) on the fundamental domain F′_{t}. Here we refer to Figure 9 where A ∪ B is a standard fundamental domain for the action of P SL_{2}(ℤ) on the upper half plane ℍ^{2}. Then G ∪ A is also a fundamental domain. We let F′_{t} = A ∪ C ∪ D ∪ E ∪ F ∪ G. One easily sees that F′_{t} is a fundamental domain for J′_{3}. Note that F_{t} and F′_{t} are homeomorphic and so there is a homeomorphism f_{t} : V_{t} ∩ ℝ_{>2}^{3} → ℍ^{2} such that f_{t}(F_{t}) = F′_{t}.

Using f_{t} we can pull back the hyperbolic metric from ℍ^{2} to V_{t} ∩ ℝ_{>2}^{3} so as to get:

Corollary 11.10. There is a hyperbolic metric on V_{t} ∩ ℝ_{>2}^{3}, t < 0, such that the action of J_{3} on V_{t} ∩ ℝ_{>2}^{3} is by hyperbolic isometries. Further, the curves defining F_{t} are geodesics in this metric. □

Remark 11.11. The strata referred to in Theorem 4 are more easily understood for n = 3, 4: Consider T̄_{0,3,1} ⊆ ℝ^{3}. This has dimension 3 and is a union of 2-dimensional strata coming from the B_{3}-invariant level sets of c′_{31} = a_{21}^{2} + a_{31}^{2} + a_{32}^{2} − a_{21}a_{31}a_{32} + 3.

For n = 4 we note that T̄_{0,4,1} ⊆ ℝ^{6} has dimension 5, however we have two independent B_{4}-invariant functions

c′_{41} = a_{21}a_{32}a_{41}a_{43} − a_{21}a_{31}a_{32} − a_{21}a_{41}a_{42} − a_{31}a_{41}a_{43} − a_{32}a_{42}a_{43}
+ a_{21}^{2} + a_{31}^{2} + a_{32}^{2} + a_{41}^{2} + a_{42}^{2} + a_{43}^{2} − 4,
c′_{42} = a_{21}^{2}a_{43}^{2} − 2a_{21}a_{31}a_{42}a_{43} + a_{31}^{2}a_{42}^{2} − 2a_{31}a_{32}a_{41}a_{42} + a_{32}^{2}a_{41}^{2} + 2a_{21}a_{31}a_{32}

+ 2a_{21}a_{41}a_{42} + 2a_{31}a_{41}a_{43} + 2a_{32}a_{42}a_{43} − 2(a_{21}^{2} + a_{31}^{2} + a_{32}^{2} + a_{41}^{2} + a_{42}^{2} + a_{43}^{2}) + 6,

which thus shows that T̄_{0,4,1} is a union of at most 4-dimensional B_{4}-invariant strata.

We now consider the cases n > 4; we use the notation of Figure 5 at the beginning of this section. We will show that there is a 1-parameter family of surfaces which give points of T_{0,n,1} where c′_{n n−1} is not constant. This will prove the last part of Theorem 4.

For n > 4 let us define the following matrices:

p_{1} = ( \begin{matrix}−1 & 0 \\ −2 & −1 \end{matrix} ) , p_{2} = ( \begin{matrix}−4 & 3 \\ −3 & 2 \end{matrix} ) ,

p_{3} = ( \begin{matrix}−1 & 2n + 3 \\ 0 & −1 \end{matrix} ) , p_{4} = ( \begin{matrix}6n − 1 & 12n^{2} \\ −3 & −6n − 1 \end{matrix} )

Now for 4 < m < n we let

p_{m} = ( \begin{matrix}8n − 4m + 15 & 4(2n − m + 4)^{2} \\ −4 & 4m − 8n − 17 \end{matrix} ) .

Finally we let

p_{n} = ( \begin{matrix}\frac{(3n+11)x_{1}+4n+15}{x_{1}+1} & \frac{(3x_{1}+4)(n+4)^{2}}{x_{1}+1} \\ \frac{−3x_{1}−4}{x_{1}+1} & \frac{−(3n+13)x_{1}−4n−17}{x_{1}+1} \end{matrix} ) .

One checks that p_{1}, …, p_{n} are all parabolics which (respectively) fix the points

f_{1} = 0, f_{2} = 1, f_{3} = ∞, f_{4} = −2n, f_{5} = −2n + 1,
f_{6} = −2n + 2, …, f_{n−1} = −(n + 5), f_{n} = −(n + 4) + x_{1}.

We will also let f_{n+1} = −(n + 3). The matrices s_{i} are as follows:

s_{1} = ( \begin{matrix}−1 & 0 \\ 2 & −1 \end{matrix} ) , s_{2} = ( \begin{matrix}−2 & 3 \\ 1 & −2 \end{matrix} ) , s_{3} = ( \begin{matrix}2 & 4n + 3 \\ −1 & −2n − 1 \end{matrix} ) ,

s_{4} = ( \begin{matrix}7 & 14n − 3 \\ −2 & 1 − 4n \end{matrix} ) ,

s_{m} = ( \begin{matrix}2m − 1 & (4m − 2)n − (2m^{2} − 8m + 3) \\ −2 & (2m − 7) − 4n \end{matrix} ) , for 4 < m < n,

s_{n} = ( \begin{matrix}\frac{nx_{1}+2n−1}{x_{1}+1} & \frac{(n^{2}+3n+1)x_{1}+2n^{2}+6n−3}{x_{1}+1} \\ \frac{−(x_{1}+2)}{x_{1}+1} & \frac{(n+3)x_{1}−2n−7}{x_{1}+1} \end{matrix} ) .

Now s_{i}(f_{i}) = g_{i} for i = 1, …, n where we put g_{i} = −(i − 1) for i = 1, …, n + 1. We also have s_{n}(f_{n+1}) = g_{n+1}. Thus following the argument at the beginning of this section we see that for sufficiently small values of x_{1} these matrices do give a point of T_{0,n,1}. Now to consider the corresponding points in T̄_{0,n,1} one solves the equations trace(p_{i}p_{j}) − (2 − a_{ij}^{2}). We now list the values of 2 − trace(p_{i}p_{j}); here we assume that 4 < m, m′ < n:

2 − trace(p_{1}p_{2}) = 6, 2 − trace(p_{1}p_{3}) = 2(2n + 3), 2 − trace(p_{1}p_{4}) = 24n^{2},

2 − trace(p_{1}p_{m}) = 8(m − 22)^{2}, 2 − trace(p_{1}p_{n}) = \frac{6(n + 4)^{2}(x_{1} + 4/3)}{x_{1} + 1},

2 − trace(p_{2}p_{3}) = 3(2n + 3), 2 − trace(p_{2}p_{4}) = (6n + 3)^{2},

2 − trace(p_{2}p_{m}) = 12(m − 23)^{2}, 2 − trace(p_{2}p_{n}) = \frac{9(n + 5)^{2}(x_{1} + 4/3)}{x_{1} + 1},

2 − trace(p_{3}p_{4}) = 3(2n + 3), 2 − trace(p_{3}p_{m}) = 4(2n + 3),

2 − trace(p_{3}p_{n}) = \frac{3(2n + 3)(x_{1} + 4/3)}{x_{1} + 1}, 2 − trace(p_{4}p_{m}) = 12(2n + m − 22)^{2},

2 − trace(p_{4}p_{n}) = \frac{9(n − 4)^{2}(x_{1} + 4/3)}{x_{1} + 1}, 2 − trace(p_{m}p_{m′}) = 16(m − m′)^{2},

2 − trace(p_{m}p_{n}) = \frac{12(m + n − 16)^{2}(x_{1} + 4/3)}{x_{1} + 1}.  (11.7)

Lemma 11.12. For n > 1 we have

−c′_{n n−1} =T race(T_{1}T_{2} … T_{n}) = n − ∑_{i_{1}<i_{2}} a_{i_{2}i_{1}}^{2} + ∑_{i_{1}<i_{2}<i_{3}} a_{i_{2}i_{1}}a_{i_{3}i_{1}}a_{i_{3}i_{2}}

− ∑_{i_{1}<i_{2}<i_{3}<i_{4}} a_{i_{2}i_{1}}a_{i_{3}i_{2}}a_{i_{4}i_{1}}a_{i_{4}i_{3}} + ∑_{i_{1}<i_{2}<i_{3}<i_{4}<i_{5}} a_{i_{2}i_{1}}a_{i_{3}i_{2}}a_{i_{4}i_{3}}a_{i_{5}i_{1}}a_{i_{5}i_{4}} − …

Proof. This is proved by induction on n, or one can use Lemma 3.1. □

Lemma 11.13. If we solve the equations 2 − trace(p_{i}p_{j}) = a_{ij}^{2} for the a_{ij} and substitute into the cycle c = a_{i_{2}i_{1}}a_{i_{3}i_{2}}a_{i_{4}i_{3}} … a_{i_{r}i_{r−1}}a_{i_{r}i_{1}} with i_{1} < i_{2} < ⋯ < i_{r} < n, then we get an integer. If we substitute into the cycle c = a_{i_{2}i_{1}}a_{i_{3}i_{2}}a_{i_{4}i_{3}} … a_{i_{r}i_{r−1}}a_{ni_{r}}a_{ni_{1}} with i_{1} < i_{2} < ⋯ < i_{r} < n, then we get an integral multiple of \frac{x_{1}+4/3}{x_{1}+1}.

Proof. Since for 4 < m ≠ m′ < n we have that 2 − trace(p_{m}p_{m′}) is a perfect square we may assume that in any such cycle we have at most one index m with 4 < m < n. This reduces the proof to checking a finite number of the remaining cases, which one does. □

We will use the above two results to show that when one substitutes any solution to the equations 2 − trace(p_{i}p_{j}) = a_{ij}^{2} into c′_{n n−1}, then the result is a non-constant function of x_{1}. In fact:

Lemma 11.14. If we solve the equations 2 − trace(p_{i}p_{j}) = a_{ij}^{2} for the a_{ij} and substitute into c′_{n n−1}, then we get a function of the form

q_{1} \frac{x_{1} + 4/3}{x_{1} + 1} + q_{2},

where q_{1}, q_{2} are integers with q_{1} being odd.

Proof. Of course there are many solutions to the equations 2 − trace(p_{i}p_{j}) = a_{ij}^{2}, however, they all differ by various signs and since we are only interested in the parity of the integer q_{1}, the specific choice of signs will not concern us. Now given Lemmas 11.12 and 11.13 the only thing we need to do is to show that q_{1} is odd. There are two cases: n even or odd. We will do the n odd case, the other being similar. Assume that n is odd. Then c′_{n n−1} is given as a sum of sums by Lemma 11.12. We look at each of these sums.

First for ∑_{i_{1}<i_{2}} a_{i_{2}i_{1}}^{2}. Note that here we are only interested in summing over those i_{1} < i_{2} with i_{2} = n. Now by (11.7) we see that a_{n1}^{2} is always even; that a_{n2}^{2} is even since n is odd; that a_{n3}^{2} is always odd; that a_{n4}^{2} is odd since n is odd; that a_{nm}^{2} is even for all 4 < m < n. Thus ∑_{i_{1}<i_{2}} a_{i_{2}i_{1}}^{2} is even.

Next for ∑_{i_{1}<i_{2}<i_{3}} a_{i_{2}i_{1}}a_{i_{3}i_{1}}a_{i_{3}i_{2}} we again need only consider the cases where i_{3} = n. Next note that if i_{1} = 1, then a_{i_{2}i_{1}}a_{ni_{1}}a_{ni_{2}} is always even. Thus we may assume that i_{1} > 1. Similarly, if i_{1} = 2, then a_{i_{2}i_{1}}a_{ni_{1}}a_{ni_{2}} is even. Further, if 4 < m < n and i_{2} = m, then a_{i_{2}i_{1}}a_{ni_{1}}a_{ni_{2}} is even. Thus we have reduced to the case i_{1} = 3, i_{2} = 4, and we find that this is odd. Thus ∑_{i_{1}<i_{2}<n} a_{i_{2}i_{1}}a_{ni_{1}}a_{ni_{2}} is odd.

For the case ∑_{i_{1}<i_{2}<i_{3}<i_{4}} a_{i_{2}i_{1}}a_{i_{3}i_{2}}a_{i_{4}i_{1}}a_{i_{4}i_{3}} we again have i_{4} = n and as above we must have i_{1} > 2. But this forces i_{3} > 4 which gives an even number also.

The rest of the cases are similar to the last one. This proves the Lemma and the last part of Theorem 4. □

Let π : B_{3}/Z(B_{3}) → P SL_{2}(ℤ) be the isomorphism so that

π(σ_{1}) = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and π(σ_{2}) = ( \begin{matrix}1 & 0 \\ −1 & 1 \end{matrix} ) ,

with these matrices acting as linear fractional transformations of the upper half plane ℍ^{2}. We now return to the situation of Theorem 11.8 and Corollary 11.10. These show that for all t < 0 there is a diffeomorphism f_{t} : V_{t} ∩ ℝ_{>2}^{3} → ℍ^{2} such that for all α ∈ B_{3}/Z(B_{3}) we have f_{t}(F_{t}) = F′_{t} and

f_{t}(α(x, y, z)) = π(α)f_{t}(x, y, z), for all (x, y, z) ∈ V_{t} ∩ ℝ_{>2}^{3}.

Now let g : ℍ^{2} → ℂ be a modular form of weight k, so that g(β(z)) = (cz + d)^{k}g(z)

for all z ∈ ℍ^{2} and β = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ P SL_{2}(ℤ) [Ko]. Then we can define a modular form on

⋃_{t<0} V_{t} ∩ ℝ_{>2}^{3} by

ḡ(x, y, z) = g(f_{t}(x, y, z)), for (x, y, z) ∈ V_{t} ∩ ℝ_{>2}^{3}.

Then we have

Theorem 11.15. For α ∈ B_{3}/Z(B_{3}), (x, y, z) ∈ V_{t} ∩ ℝ_{>2}^{3} and a modular form g of weight k we have

ḡ(α(x, y, z)) = (cz + d)^{k}ḡ(x, y, z), where π(α) = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

Proof. For (x, y, z) ∈ V_{t} ∩ ℝ_{>2}^{3} we have:

ḡ(α(x, y, z)) = g(f_{t}α(x, y, z))
= g(π(α)f_{t}(x, y, z))
= (cz + d)^{k}g(f_{t}(x, y, z))
= (cz + d)^{k}ḡ(x, y, z),

where π(α) = ( \begin{matrix}a & b \\ c & d \end{matrix} ) . □

Since T_{0,3,1} can be thought of as a subset of ⋃_{t<0} V_{t} ∩ ℝ_{>2}^{3} we see that the above result gives a way of defining modular forms on T_{0,3,1}. It would be nice to have an explicit formula for the functions f_{t}.

Acknowledgement. The author wishes to thank Jim Cannon, Tyler Jarvis, Ken Johnson and Eric Swenson for useful conversations regarding this paper.

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