A strong Schottky Lemma for nonpositively curved singular spaces

To John Stallings on his sixty-fifth birthday

Roger C. Alperin, Benson Farb^{∗}and Guennadi A. Noskov^{†}

January 22, 2001

1 Introduction

The classical Schottky Lemma (due to Poincare, Klein, Schottky) gives a criterion for a pair of isometries g, h of hyperbolic space to have powers g^{m}, h^{n} which generate a free group. This criterion was generalized by Tits to pairs of elements in linear groups in his proof of the Tits alternative.

In this paper we give a criterion (Theorem 1.1) for pairs of isometries of a nonpositively curved metric space (in the sense of Alexandrov) to generate a free group without having to take powers. This criterion holds only in singular spaces, for example in Euclidean buildings; in fact our criterion takes a particularly simple form in that case (Corollary 1.2).

The original motivation for our criterion was to prove that the four dimensional Burau representation is faithful. While linearity of braid groups is now known, this question is still open; it is well-known to be related to detecting the unknot with the Jones polynomial . It was shown in [4] that the faithfulness question is equivalent to proving that a specific pair of elements in GL_{3}(Z[t, t^{−1}]) generate a free group. In §4 we show that these elements are the image of the free group F_{2} under a representation ρ_{0,0} lying in a 2-parameter family of representations

∗Supported in part by the NSF and by the Alfred P. Sloan Foundation.

†Supported by GIF-grant G-454-213.06/95

ρ_{α,β} : F_{2} → GL_{3}(Q(t))

for α, β ∈ Q. Our criterion applies to give faithfulness of ρ_{α,β} for all α, β except, unfortunately, for α, β ∈ {0, 1}.

The criterion. For definitions of the terms we use we refer the reader to §2. A complete, nonpositively curved (i.e. CAT(0)) metric space X is said to have no fake angles if there is no pair of geodesics which issue from the same point, have zero angle at that point, yet are disjoint except at that point. This is a very weak condition, and is satisfied for example by any piecewise Euclidean CAT(0) simplicial complex with finitely many isometry types of cells (Proposition 2.1).

Theorem 1.1 (Strong Schottky). Let X be a complete CAT(0)-space with no fake zero angles. Let g_{1}, g_{2} be isometries of X of hyperbolic type with axes A_{1}, A_{2} respectively. Assume that either one of the following holds:

1. S = A_{1} ∩ A_{2} is a bounded segment. Let s_{−}, s_{+} be its endpoints and let A_{1}^{±}, A_{2}^{±} be the infinite rays of A_{1}, A_{2} with the origins s_{−}, s_{+}. Assume that the angles between A_{1}^{+}, A_{2}^{+} and between A_{1}^{−}, A_{2}^{−} are equal to π, and that the translation lengths of g_{1}, g_{2} are strictly greater than the length of S.

2. A_{1}, A_{2} are disjoint, and there is a geodesic B connecting a_{1} ∈ A_{1} to a_{2} ∈ A_{2} with all four angles between B and A_{1}, A_{2} at a_{1}, a_{2} equal to π.

Then the group generated by g_{1}, g_{2} is free.

The case of buildings. Let X be a thick affine building. For example, let X = T_{ν} be the building associated to GL_{n}(K), n ≥ 2 where K is a discretely-valued field with valuation ν. The link link(x) of a point x ∈ X is a spherical building. Two chambers in the link are called opposite if their distance is maximal in the link. Any chamber in the link has a unique opposite chamber. We say that two chambers in A are opposite if their intersections with the link of a common intersection point are opposite.

An isometry f ∈ Isom(X) of hyperbolic type is said to be generic if none of its (parallel) axes are contained in any wall of any apartment of X. Note that f is generic if and only if it has a unique invariant apartment P_{f}. The axis of f is contained in the union of two (equivalence classes of) sectors P_{f}^{+}, P_{f}^{−} of P_{f} which are invariant by translation by f, respectively f^{−1}. A generic isometry f determines, for any fixed choice of basepoint x ∈ P_{f}, a pair of chambers in link(x). We say that generic f, g ∈ Isom(X) are opposite if P_{f} ∩ P_{g} = x ∈ X and if each of the chambers determined by f is opposite in link(x) to each of the chambers determined by g.

As a straightforward consequence of Theorem 1.1 we have the following.

Corollary 1.2 (Strong Schottky for buildings). Let X be a (thick) Euclidean building, and let f, g ∈ Isom(X). If f, g are opposite then they generate a free subgroup of Isom(X).

In §4 we will apply Corollary 1.2 to a 2-parameter family of pairs of elements in the affine building associated to GL_{3}(Q(t)).

2 CAT(0) preliminaries

2.1 Definitions

Let X be a geodesic metric space. The comparison triangle for a geodesic triangle Δ in X is the Euclidean triangle Δ′ with the same side lengths as Δ; this triangle is unique up to isometry. We say that X is CAT (0), or nonpositively curved, if for any geodesic triangle Δ in X and any two points x, y on Δ, the distance between x and y in X is less than or equal to the Euclidean distance between the corresponding points x′, y′ on the comparison triangle Δ′ ( [3], II.1.1). The CAT(0) condition implies the uniqueness of geodesics and the geodesicity of local geodesics. In the following we assume that X is a complete CAT(0) space.

Let ε > 0 and let σ_{1}, σ_{2} : [0, ε] → X be two unit speed geodesics with σ_{1}(0) = σ_{2}(0) =: x. For s, t ∈ (0, ε) let γ(s, t) be the angle at x′ of the comparison triangle x′σ_{1}(s)′σ_{2}(t)′. Then γ(s, t) is monotonically decreasing as s, t decrease and hence ∠(σ_{1}, σ_{2}) := lim_{s,t→0} γ(s, t) exists and is called the angle subtended by σ_{1} and σ_{2} ([1], I.(3.8)). A metric expression for the angle is given by the “cosine theorem”([1], I.(3.11)

cos(∠(σ_{1}, σ_{2})) = lim_{s,t→0} \frac{s^{2} + t^{2} − d^{2}(σ_{1}(s), σ_{2}(t))}{2st}.

It follows from this formula that if the angle is strictly less than π/2, and s/t is sufficiently small, then d(σ_{1}(s), σ_{2}(t)) < t, in particular σ_{1}(0) = σ_{2}(0) does not minimize the distance from σ_{2}(t) to the segment σ_{1}([0, ε]). We need this for the properties of the projection map below.

The sum of the angles of a geodesic triangle is less than or equal to π ( [1], I.5.2). It follows immediately from this property that if I, J are geodesic segments issuing from the same point and the subtended angle equals π then the concatenation of these segments is also a geodesic segment. If σ is a geodesic segment (possibly infinite) then for any x ∈ X there is ([1], I.5.6) a unique point p_{σ}(x) ∈ σ such that d(x, p_{σ}(x)) = d(x, σ). The map p_{σ} is called the projection onto σ. It follows from the cosine formula and the remarks above that for each x ∈ X, the angles of [x, p_{σ}(x)] with σ both are greater than or equal to π/2.

2.2 Fake zero angles

We say that a complete CAT(0) space X has fake zero angles if there are two geodesics issuing from the same point, are disjoint except at that point, and if the angle subtended by them at that point is zero.

Proposition 2.1. A piecewise Euclidean CAT(0) complex X with finitely many isometry types of cells has no fake zero angles.

Proof. The assumptions imply that the path metric on X is geodesic and complete. The angles can be defined in terms of link distance [2]. Namely, let X be a piecewise Euclidean complex, x ∈ X. The link Lk_{x} A of the Euclidean cell A is the set of unit tangent vectors ξ at x such that a nontrivial line segment with initial direction ξ is contained in A. We define the link link(x) of x ∈ X by link(x) = ∪_{A∋x} Lk_{x} A, where the union is taken over all closed cells containing x. Angles in Lk_{x} A induce a natural length metric d_{x} on link(x) which turns it into a piecewise spherical complex. The angle between ξ, η ∈ link(x) is then defined by ∠_{x}(ξ, η) = min(d_{x}(ξ, η), π).

Any two segments σ_{1}, σ_{2} in X with the same endpoint x have the natural projection image in the link of x and ∠_{x}(σ_{1}, σ_{2}) equals the angle between these two projections. Now the assertion of the lemma is clear since if the segments are disjoint, apart from the origin, then their images in the link are distinct and hence the link distance is nonzero. ⋄

Example (V. Berestovskii): Take R^{2} with the positive x-axis removed and glue in the region {(x, y) : x ≥ 0, y ≤ x^{2}} along the obvious isometry of the boundary. The result is a CAT(0) space with fake zero angles.

3 Proof of Theorem 1.1

We will need the following well-known lemma.

Lemma 3.1 (Ping-Pong Lemma). Let Γ be a group of permutations on a set X, let g_{1}, g_{2} be elements of Γ of order at least 3. If X_{1}, X_{2} are disjoint subsets of X and if, for all n ≠ 0, i ≠ j, g_{i}^{n}X_{j} ⊂ X_{i} then g_{1}, g_{2} freely generate the free group F_{2}.

The proof of Theorem 1.1 divides into two cases, depending on which of the two hypotheses is assumed.

Assuming (1): Let s be the midpoint of S. Let D_{1}, D_{2} be the fundamental domains for the action of g_{1}, g_{2} on A_{1}, A_{2}, chosen as open segments on A_{1}, A_{2} with the center s. Let p_{1}, p_{2} be the geodesic projection maps of X onto A_{1}, A_{2} respectively. Set X_{1} = p_{1}^{−1}(A_{1} − D_{1}) and X_{2} = p_{2}^{−1}(A_{2} − D_{2}). To apply the Ping-Pong Lemma we need to show that X_{1} ∩ X_{2} is empty. Suppose, to the contrary, it is not and let x ∈ X_{1} ∩ X_{2}. So p_{1}(x) ∈ A_{1}, p_{2}(x) ∈ A_{2}. Suppose that p_{1}(x) ∈ A_{1}^{+}, p_{2}(x) ∈ A_{2}^{−}, the other cases being similar. Consider the geodesic triangle xp_{1}(x)p_{2}(x). The sum of its angles is at most π and we would like to get the contradiction with this. Suppose first that the angle at x is nonzero. By the angles assumption we get that [p_{1}(x), p_{2}(x)] is the concatenation of [p_{1}(x), s_{−}], [s_{−}, s_{+}], [s_{+}, p_{2}(x)]. By the property of the projection map, the angle of the triangle at p_{1}(x), p_{2}(x) is at least π/2. Hence the sum of the angles is strictly greater than π.

Suppose now that the angle at x is zero. Then by the fake zero angles assumption, the geodesics (x, p_{1}(x)], (x, p_{2}(x)] are not disjoint hence [x, p_{1}(x)]∩[x, p_{2}(x)] = [x, y] for some y ≠ x. To get the contradiction consider the triangle yp_{1}(x)p_{2}(x). If y is strictly closer to x than both of p_{1}(x), p_{2}(x), then the y-angle of the triangle yp_{1}(x)p_{2}(x) is nonzero and the sum of angles in this triangle is strictly greater than π - a contradiction. If not, then say y = p_{1}(x). Since y lies on [x, p_{2}(x)], its projection onto A_{2} is the same as that of x. Hence p_{2}(x) = p_{2}(y). By definition, p_{2}(y) is the point on A_{2} closest to y. But s_{+} ∈ A_{2} lies on the geodesic [y, p_{2}(x)] and s_{+} ≠ p_{2}(x); again, this is a contradiction.

Finally it remains to check that g_{i}^{n}X_{j} ⊂ X_{i}, i ≠ j, n ≠ 0. Note first that g_{i} commutes with p_{i}. Indeed, for any x ∈ X, the point p_{i}(x) is the unique point in A_{i} such that d(x, p_{i}(x)) = d(x, A_{i}). But

d(g_{i}x, A_{i}) = d(g_{i}x, g_{i}A_{i}) = d(x, A_{i}) = d(x, p_{i}(x)) = d(g_{i}x, g_{i}p_{i}(x))

That is, the point g_{i}p_{i}(x) realizes the distance d(g_{i}x, A_{i}) and thus it is the projection of g_{i}x hence g_{i}p_{i}(x) = p_{i}g_{i}(x).

Clearly g_{i}^{n}D_{i} ⊂ A_{i} − D_{i}, n ≠ 0, whence g_{i}^{n}(X − X_{i}) ⊂ X_{i}. Finally, X_{j} ⊂ X − X_{i}, hence g_{i}^{n}(X_{j}) ⊂ X_{i}.

Assuming (2): Let p_{1}, p_{2} be the geodesic projection maps of X onto A_{1}, A_{2} respectively. Let D_{1}, D_{2} be the fundamental domains for g_{1}, g_{2} chosen as the open segments on A_{1}, A_{2} with the centers a_{1}, a_{2} respectively. Set X_{1} = p_{1}^{−1}(A_{1} − D_{1}) and X_{2} = p_{2}^{−1}(A_{2} − D_{2}) and repeat the argument of the first case. ⋄

4 A family of examples

4.1 Motivation: the Burau representation

The braid group on n strands, denoted B_{n}, is the group with generators s_{1}, s_{2}, …, s_{n−1} and relations

s_{j}s_{k} = s_{k}s_{j}, s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1}

for all possible i, j, k with |k − j| ≥ 2 and i + 1, j, k ≤ n − 1. The Burau representation s_{i} ↦ σ_{i} is a natural representation of B_{n} on the n-dimensional linear space V = K^{n} over the field K = Q(t); in the standard basis {e_{i}|i = 1, …, n} the Burau representation is determined by

σ_{i}(e_{j}) = \{ \begin{matrix}(1 − t)e_{i} + e_{i+1} & if j = i \\ te_{i} & if j = i + 1 \\ e_{j} & if j ≠ i, i + 1 \end{matrix}

The subspace Ke, where e=∑_{i=1}^{n} e_{i}, is clearly an invariant subspace for the group action, and the group acts trivially there. Thus the quotient space V/Ke has a natural action of the braid group. In the induced basis e_{1}, e_{2}, …, e_{n−1} the elements σ_{i} act via matrices which we denote by b_{1}, b_{2}, …, b_{n−1}. The only new aspect in this reduced Burau representation is that

b_{n−1}(e_{n−1}) = (1 − t)e_{n−1} + e_{n} = −(∑_{i=1}^{n−2} e_{i} + te_{n−1}).

Our interest is in the case of four strands. In this case there is the following well known result (however, the matrices are incorrectly specified in [4]).

Proposition 4.1 ([4], Theorem 3.19). The reduced representation of B_{4} is faithful iff it is faithful on the free group generated by a = s_{3}s_{1}^{−1} and b = s_{2}(s_{3}s_{1}^{−1})s_{2}^{−1}.

Let f (resp. k) be the image of a (resp. b) in GL_{3}(K) under the reduced Burau representation. It is not difficult to see that both f and k are diagonalizable. In fact, by conjugating the Burau representation, and changing t to −t, we may take f to be the diagonal matrix with entries 1, t^{−1}, t, and k becomes the matrix k = sfs^{−1} where

s = (1 − t)^{−2} ( \begin{matrix}−(1 + t) & 1 + t^{2} & −t(1 + t^{2}) \\ 1 & −t & t \\ 1 & −1 & t^{2} \end{matrix} )

We consider the action of GL_{3}(K) on the Bruhat-Tits building T_{ν} for the field K with the discrete valuation ν = ν_{∞} at infinity. We have two elements of GL_{3}(K) which are acting so that they each stabilize an entire apartment of T ; these apartments A_{u} and A_{v} on general principles will meet in a convex subset of the building.

However, by analogy with the case of actions on trees, we might expect that if the intersection of P_{f} and P_{k} is sufficiently small with respect to the translation distances of f and k then the group generated by f and k is free. Since f is semisimple, it is easy to see that it acts by translation on its apartment, and we will see that no axis for f is parallel to any wall in the apartment; similarly for k. We can determine the intersection P_{f} ∩ P_{k}.

Lemma 4.2 (P_{f} ∩ P_{k} is a point). The intersection of P_{f} and P_{k} consists of precisely one lattice class.

Proof. Let ν denote the discrete valuation, with valuation ring O and uniformizer π. The lattices which represent the lattice classes in P_{f} are

L_{a_{1},a_{2},a_{3}} = Oπ^{a_{1}}e_{1} + Oπ^{a_{2}}e_{2} + Oπ^{a_{3}}e_{3}.

Since k = sfs^{−1}, the lattices in P_{k} are precisely the lattice classes [sL] for [L] in P_{f}, the standard apartment. Thus a lattice class of L_{−a_{1},−a_{2},−a_{3}} belongs to the intersection if there are integers a_{1}, a_{2}, a_{3} and b_{1}, b_{2}, b_{3} so that [sL_{b_{1},b_{2},b_{3}}] = [L_{−a_{1},−a_{2},−a_{3}}]. In other words, there is some matrix m in GL_{3}(O) so that

s ( \begin{matrix}π^{b_{1}} & 0 & 0 \\ 0 & π^{b_{2}} & 0 \\ 0 & 0 & π^{b_{3}} \end{matrix} ) = ( \begin{matrix}π^{−a_{1}} & 0 & 0 \\ 0 & π^{−a_{2}} & 0 \\ 0 & 0 & π^{−a_{3}} \end{matrix} ) m

This gives rise to the conditions: π^{a_{i}}s_{i,j}π^{b_{j}} ∈ O for all entries s_{i,j}, 1 ≤ i, j ≤ 3 of s. This implies

a_{i} + b_{j} + ν(s_{i,j}) ≥ 0

Since m is invertible, ν(det(m)) = 0. This implies

a_{1} + a_{2} + a_{3} + b_{1} + b_{2} + b_{3} + νdet(s) = 0.

We can convert many of the above inequalities into equalities by the following argument. Direct calculation gives ν(det(s)) = 2. Hence

2 = (a_{2} + b_{1}) + (a_{1} + b_{2}) + (a_{3} + b_{3}) ≥ 2 + 0 + 0

and therefore a_{2} + b_{1} = 2, a_{1} + b_{2} = 0, and a_{3} + b_{3} = 0. It follows similarly that a_{1}+b_{1} = 1, a_{2}+b_{2} = 1, a_{3}+b_{1} = 2, a_{1}+b_{3} = −1, a_{2}+b_{3} = 0, a_{3}+b_{2} = 1. From this we can immediately solve the nine equations in six variables to get the solutions a_{1} = −1 − c, a_{2} = −c, a_{3} = −c, b_{1} = 2 + c, b_{2} = 1 + c, b_{3} = c and so this determines exactly one lattice class solution to the intersection of the two apartments. ⋄

4.2 A 2-parameter family of representations

Based on the example afforded by the Burau representation, we consider the transformations f and its conjugate k = sfs^{−1}, where

s = (1 − t)^{−2} ( \begin{matrix}−(1 + t) & 1 + t^{2} & −t(1 + t^{2}) \\ 1 & −t & t + βt^{2} \\ 1 & −1 + αt & t^{2} \end{matrix} )

for α and β any rational numbers. We can think of this as giving a 2parameter family of representations

ρ_{α,β} : F_{2} → GL_{3}(Q(t))

for α, β ∈ Q. Consider any fixed parameters, giving a pair f, k. As in the proof of Lemma 4.2, the invariant apartments of f and k meet at exactly one point.

The lattices which represent the lattice classes in the standard apartment A are L_{a_{1},a_{2},a_{3}} = Oπ^{a_{1}}e_{1} + Oπ^{a_{2}}e_{2} + Oπ^{a_{3}}e_{3}, π represents the uniformizer t^{−1}. This apartment is stabilized by f. In this standard basis, e_{1}, e_{2}, e_{3}, f is represented by a diagonal matrix with diagonal entries 1, t^{−1}, t. The apartment stabilized by k = sfs^{−1} is sA. We consider the lattice class x of the lattice L_{−1,0,0} in the standard apartment as the special vertex which is the common cone point of our sectors P_{f}^{+}, P_{f}^{−}, P_{k}^{+}, P_{k}^{−}. This vertex is also sL_{−2,−1,0} by the calculation above. The walls of the sectors P_{f}^{+} and P_{f}^{−} are then easily seen to be the subcomplex represented by the lattice classes of M_{n} = L_{−1,n,0} n ∈ Z, and N_{n} = L_{−1,0,n} n ∈ Z. The translation f takes [L_{−1,0,0}] to [L_{−1,1,−1}] with axis bisecting the walls of the sector. We can apply s to this configuration to obtain a similar description for the axis and sectors for the element k.

We shall calculate the link of the vertex x and represent it in terms of the spherical building of GL_{3}(Q), where the rational field Q is the residue field of the field Q(t) with respect to the valuation at infinity. A q-simplex in the building is represented by a chain of lattices L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{q} with πL_{q} ⊂ L_{0} and L_{r+1}/L_{r} ≅ Q, r ≥ 0. The link of the vertex [L] is the simplicial complex whose vertices are lattice classes [L′] so that πL ⊂ L′ ⊂ L. By taking chains of such classes we obtain a simplex in the spherical building of GL_{3}(Q), viewed as flags in the 3-dimensional Q-vector space, L/πL.

With the labelling of the walls as above, the first chamber in the sector P_{f}^{+} is represented by the lattice classes of L = L_{−1,0,0} and the lattice classes of M_{1} = L_{−1,1,0} and N_{−1} = L_{−1,0,−1} while the first chamber in P_{f}^{−} is represented by the lattice classes of L_{−1,0,0} and the lattice classes of M_{−1} = L_{−1,−1,0} and N_{1} = L_{−1,0,1}. We can apply s to these classes to represent the other leading chambers in P_{k}^{+} and P_{k}^{−}.

In order to obtain the precise configurations in P_{f}^{+}, consider the leading chamber which contains the lattice classes of M_{1} and N_{−1}. We have the chain of lattices πL ⊂ πN_{−1} ⊂ M_{1} ⊂ L, so in link(x) we have the flag of subspaces

X_{1} = πN_{−1}/πL ⊂ X_{2} = M_{1}/πL,

which is the line X_{1} with basis e_{3} as a subspace of X_{2} with basis {e_{1}, e_{3}}. The same can be done for the leading chamber of P_{f}^{−} to obtain the flag

πL ⊂ πM_{−1} ⊂ N_{1} ⊂ L

giving the line Y_{1} with basis e_{2} as a subspace of Y_{2} with basis {e_{1}, e_{2}}. It is immediate that the edge X_{1} ⊂ X_{2} is opposite to Y_{1} ⊂ Y_{2}.

Similarly, we have the flags s(X_{1}) ⊂ s(X_{2}) and s(Y_{1}) ⊂ s(Y_{2}) in link(x) coming from the sectors P_{k}^{+} and P_{k}^{−} based at the point [s(L_{−2,−1,0})] = t, since [L_{−1,0,0}] = [s(L_{−2,−1,0})]. Consider the change of basis matrix m (as described in the proof of Lemma 4.2) and reduce mod π = \frac{1}{t} to obtain the rational matrix

( \begin{matrix}−1 & 1 & −1 \\ 1 & −1 & β \\ 1 & α & 1 \end{matrix} )

to find that s(X_{1}) has basis (−1, β, 1) and s(X_{2}) has basis {(−1, 1, 1), (−1, β, 1)}, s(Y_{1}) has basis (1, −1, α) and s(Y_{2}) has basis {(1, −1, α), (−1, β, 1)}. Thus these edges are opposite to the edge X_{1} ⊂ X_{2} for α and β, rationals which are neither 0 nor 1 ; but not opposite if α or β is 0 or 1.

To see this we can describe the local hexagons in the link. Oppositeness in this case means that the two 2-dimensional subspaces intersect along a line which is not the special line in each. We show that X_{1} ⊂ X_{2} is opposite to s(X_{1}) ⊂ s(X_{2}) and s(Y_{1}) ⊂ s(Y_{2}). For example, X_{2} is spanned by e_{3} and e_{1}, and sX_{2} is spanned by (−1, β, 1) and (−1, 1, 1). The subspace X_{2} ∩ s(X_{2}) does not contain either the line generated by e_{3} or the line generated by (−1, β, 1) iff β ≠ 0. Thus we have oppositeness in this case. Similarly, we can treat the cases of X_{2}, s(Y_{2}), and Y_{2}, s(Y_{2}).

References

[1] W. Ballmann, Lectures on spaces of nonpositive curvature, with an appendix by Misha Brin: Ergodicity of geodesic flows., DMV Seminar. Bd. 25., Basel: Birkhaeuser Verlag, 1995.

[2] W. Ballmann and M. Brin, Rank rigidity of Euclidean polyhedra, Amer. J. Math., 122(5):873–885, 2000.

[3] M. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Grundlehren der math. Wissen., Vol. 139, Springer-Verlag, 1999.

[4] J. Birman, Braids, links, and mapping class groups, Annals of Math. Studies, No. 82, Princeton Univ. Press, 1974.

[5] K. Brown, Buildings, Springer-Verlag, 1989.

Roger C. Alperin: Dept. of Mathematics and Computer Science, San Jose State University 1 Washington Square San Jose, CA 95192 E-mail: alperin@mathcs.sjsu.edu

Benson Farb: Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, IL 60637 E-mail: farb@math.uchicago.edu

Guennadi A. Noskov: Institute of Mathematics, Russian Academy of Sciences Pevtsova 13 Omsk 644099, RUSSIA E-mail: noskov@private.omsk.su