Journal of Lie Theory Volume 12 (2002) 259–264 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

Strong Tits Alternative for Subgroups of Coxeter Groups

Guennadi A. Noskov and Èrnest B. Vinberg

Communicated by K. H. Hofmann

Abstract. It is proved that any subgroup of a Coxeter group has a finite index subgroup which either is abelian, or has a non-abelian free quotient.

1. Introduction

A group is called large if a subgroup of finite index in it has a non-abelian free quotient. Largeness of some classes of groups was proved in [1], [7]. [6], [14], [4]. [9]. [8], [2], [10]. In particular, in [10] (see also [5]) it was proved that any non-affine infinite indecomposable Coxeter group of finite rank is large. We prove

Theorem 1.1. Any subgroup Γ of a Coxeter group W of finite rank is either large or virtually abelian.

We make use of the technique of trees of [10] and the Moussong complex of a Coxeter group [12].

The work was supported by SFB 343 of Bielefeld University. We thank this university for its hospitality. The work of the second author was also supported by RFBR grant 98-01-00598.

2. Piecewise Euclidean cell complexes

A Euclidean cell is a compact intersection of a finite number of affine half-spaces in ℝ^{n}. A piecewise Euclidean cell complex X is a connected locally finite cell complex made up by gluing together Euclidean cells via isometries of their faces. The canonical metric in each cell allows to measure the lengths of finite polygonal paths in X. The path metric d on X is defined by setting the distance between x, y ∈ X to be the infimum of the lengths of polygonal paths joining x to y. Under assumption that there are only finitely many isometry types of cells in X, the path metric is proper, that is any closed bounded subset in X is compact and, hence, there is a length minimizing path between any two points in X ([3], Chapter I.7).

Let Δ be a geodesic triangle in X. The comparison triangle for Δ is the Euclidean triangle Δ′ with the same side lengths as Δ. We say that X is

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag nonpositively curved, if for any geodesic triangle Δ in X and 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′. The nonpositive curvature condition implies the uniqueness of geodesics and the geodesicity of local geodesics.

Let X be a nonpositively curved piecewise Euclidean cell complex with finitely many isometry types of cells and g be an isometry of X without fixed points. An axis of g is a g-invariant bi-infinite geodesic (on which g acts as a non-trivial translation). The existence of an axis can be proved as follows. Let d_{g} : X → ℝ_{+} be the displacement function of g defined by d_{g}(x) = d(gx, x). Assume that it attains a (positive) minimum at some point x ∈ X. Then the union of the geodesic segments [g^{n}x, g^{n+1}x]_{n∈ℤ} is an axis of g [3].

It is easy to see that the translation length is one and the same for all axes of g [3]. It follows that, if the function d_{g} attains a minimum, any axis is obtained in the way described above. The existence of a minimum for the displacement function can be proved by means of the following

Lemma 2.1. ([3], Proposition 6.10(2)) Let Γ be a cocompact discrete group of isometries of a proper metric space X. Then, for any g ∈ Γ, the displacement function d_{g} attains a minimum.

Proof. Let |g| = inf{d_{g}(x) : x ∈ X}, and let D be a fundamental domain of Γ. Then there exist x_{n} ∈ D and g_{n} ∈ Γ (n ∈ ℕ) such that d_{g}(g_{n}x_{n}) → |g|. Since D is compact, we may assume that x_{n} tends to some x_{0} ∈ D. Then d_{g}(g_{n}x_{0}) → |g|. We have d_{g}(g_{n}x_{0}) = d(g_{n}^{−1}gg_{n}x_{0}, x_{0}). Since the points g_{n}^{−1}gg_{n}x_{0} lie in a bounded subset, there are only finitely many of them. Passing to a subsequence, we may assume that g_{n}^{−1}gg_{n}x_{0} = g_{0}^{−1}gg_{0}x_{0} for some g_{0} ∈ Γ and all n ∈ ℕ. Then d_{g}(g_{0}x_{0}) = d(g_{0}^{−1}gg_{0}x_{0}, x_{0}) = |g|, so the function d_{g} attains a minimum at g_{0}x_{0}.

3. Moussong complex of a Coxeter group

In [12] G. Moussong constructed for any Coxeter group W of finite rank a contractible piecewise Euclidean complex M_{W} of nonpositive curvature on which W acts discretely and cocompactly by isometries. Moreover, there are only finitely many isometry types of cells in M_{W}. In particular, M_{W} is proper relative to the path metric. All the cells with their Euclidean metric are isometrically embedded into M_{W} .

If W is finite, then M_{W} is just one cell, which is obtained as the convex hull P of the W -orbit of a suitable point p in the standard linear representation of W as a group generated by reflections. The point p is chosen in such a way that its stabilizer in W be trivial and all the edges of P be of length 1. The faces of P are naturally identified with the Moussong complexes of the subgroups of W conjugate to standard subgroups.

In the general case, the complex M_{W} is built up of the Moussong complexes of maximal finite subgroups of W gluing together along their faces corresponding to common finite subgroups. The 1-skeleton of M_{W} , considered as a combinatorial graph is isomorphic to the Cayley graph C_{W} of W with respect to the standard generating set.

The walls in M_{W} are the fix point sets of reflections in W. The intersections of a wall H m with cells of M_{W} supply H with a structure of a piecewise Euclidean cell complex with finitely many isometry types of cells. Any geodesic joining two points of H entirely lies in H. It follows that the induced metric of H coincides with its path metric. Hence, H is nonpositively curved. The same is true for any non-empty intersection of walls.

Lemma 3.1. Let H_{1}, ..., H_{k} be walls such that H_{1} ∩ ... ∩ H_{k} ≠ Ø and, for any i, H_{i} does not contain H_{1} ∩ ... ∩ H_{i−1}. Then the codimension of H_{1} ∩ ... ∩ H_{k} in M_{W} is at least k.

Proof. Let r_{i} be the reflection in H_{i}. It follows from the condition that r_{1}, ..., r_{k} generate a finite subgroup and, for any i, r_{i} is not contained in the subgroup generated by r_{1}, ..., r_{i−1}. Let C be the cell defined by any maximal finite subgroup of W containing r_{1}, ..., r_{k}. Then, for any i, H_{i} ∩ C does not contain H_{1} ∩ ... ∩ H_{i−1} ∩ C. Hence, the codimension of H_{1} ∩ ... ∩ H_{k} ∩ C in C is k. Since the dimension of any such C does not exceed the dimension of M_{W}, the assertion of the lemma follows.

Lemma 3.2. Let an element g ∈ W of infinite order leave invariant a nonempty intersection of walls H_{1}, ..., H_{k}. Then g has an axis lying in H_{1} ∩ ... ∩ H_{k}.

Proof. Consider an axis of the restriction of g to H_{1} ∩ ... ∩ H_{k}. It is a geodesic in M_{W} and hence an axis of g in M_{W}.

The following lemma is well-known: see, e.g., [11], Lemma 2.1.

Lemma 3.3. Let W_{0} be a normal torsionfree subgroup in W. Then for any g ∈ W_{0} and any wall H either gH = H or gH ∩ H = Ø.

Proof. Let r ∈ W be the reflection in H. Then grg^{−1} is the reflection in gH. If gH ∩ H ≠ Ø, the element grg^{−1}r fixes gH ∩ H pointwise. By the choice of W_{0} it is trivial and thus g commutes with r, so grg^{−1} = r and gH = H.

Any wall in the Moussong complex is “totally geodesic” in the following sense [13].

Lemma 3.4. Any geodesic containing at least two points of a wall H entirely lies in H.

Proof. Let two consecutive segments σ_{1}, σ_{2} of a geodesic lie in some cells C_{1}, C_{2} respectively, and their common point p lie in the intersection C_{0} of C_{1} and C_{2}. Let us prove that the orthogonal projections σ′_{1}, σ′_{2} of σ_{1}, σ_{2} (or, maybe, some smaller segments) to C_{0} continue one another. If it is not so, then moving p along the bisector of the angle formed by σ′_{1} and σ′_{2}, we obtain a shorter path which contradicts the geodesicity. Now, if σ_{1} lies on a wall H, then (since H is perpendicular to C_{0}) σ′_{1} and, hence, σ′_{2} lie on H as well. But then σ_{2} also lies on H. This proves the lemma.

Any wall divides M_{W} into two connected components [13]. All the walls yield a decomposition of M_{W} into (closed) convex sets called chambers. The set of all chambers with an appropriate adjacency relation is isomorphic to the Cayley graph C_{W} . The Cayley graph distance between two chambers equals the number of walls separating these chambers.

We say that a wall H and a bi-infinite geodesic A are transversal if H ∩ A is a one point set.

Lemma 3.5. For any bi-infinite geodesic A there is a wall, transversal to A.

Proof. Since the geodesic A cannot lie in one chamber, there are two points of it that do not belong to one chamber. These points are separated by a wall, which is thereby transversal to A by Lemma 3.4.

4. Trees

Let Γ be a subgroup in a Coxeter group W and let H be a wall in M_{W} such that for any g ∈ Γ either gH = H or gH ∩ H = Ø. The walls gH, g ∈ Γ, yield a partition of M_{W} into (closed) convex sets, which we shall call Γ-chambers. Consider the graph T = T (Γ, H), whose vertices (resp. edges) are the Γ-chambers (resp. the walls gH, g ∈ Γ) and the incidence is defined by inclusion. Clearly, T is a tree and Γ acts naturally on T . By construction this action is transitive on the set of edges. The following assertion may be applied to prove that Γ is large.

Proposition 4.1. ([10], Proposition 2) Let a group Γ act on a tree T which is not a star or a line. Suppose that Γ does not reverse edges and its action on the set of edges is transitive and residually finite. Then Γ is large.

Recall that an action of a group Γ on the set X is called residually finite, if for any different x, x′ ∈ X there exist an action of Γ on a finite set F and a Γ-equivariant map f : X → F such that f(x) ≠ f(x′). One sees easily that if an action Γ : X is residually finite then the induced actions Δ : X, where Δ is a subgroup of Γ, and Γ : Y , where Y is an invariant subset of X, are also residually finite. These properties and the fact that the action of a Coxeter group of finite rank on the set of reflections (= the action on the set of walls) is residually finite [10] imply the following

Proposition 4.2. Let Γ be a subgroup in a Coxeter group W of finite rank and let H be a wall in M_{W} such that for any g ∈ Γ either gH = H or gH ∩ H = Ø. Then the action of Γ on the set of edges of the tree T (Γ, H) is residually finite.

5. Proof of Theorem

Since the group W is linear, it contains a torsionfree normal subgroup W_{0} of finite index. Replacing Γ with Γ ∩ W_{0}, we may assume that Γ ⊂ W_{0}. Then by Lemma 3.3 for any wall H and any g ∈ Γ either gH = H or gH ∩ H = Ø.

Take any g ∈ Γ, g ≠ e. Let A be an axis of g. By Lemma 3.5 there is a wall H transversal to A. If H meets A at a point p, then the points g^{−1}p, gp and, hence, the walls g^{−1}H, gH are on different sides of H. It follows that the corresponding edges of the tree T = T (Γ, H) cannot have a common vertex. This shows that T is not a star. Possibly passing to a subgroup of index 2, we may assume that Γ does not reverse edges of T (see Lemma 3 in [10]). If T is not a line, then Γ is large by Proposition 4.1.

Let us assume now that the tree T = T (Γ, H) is a line for any wall H transversal to an axis of some element of Γ.

Take any nontrivial element g_{1} ∈ Γ. Let A_{1} be an axis of it, and H_{1} a wall transversal to A_{1}. If the stabilizer Γ_{1} of H_{1} in Γ is not trivial, take any nontrivial element g_{2} ∈ Γ_{1}. By Lemma 3.2 it has an axis A_{2} contained in H_{1}. Let H_{2} be a wall transversal to A_{2}. Obviously, H_{2} does not contain H_{1}. Let Γ_{2} be the intersection of the stabilizers of H_{1} and H_{2} in Γ. If Γ_{2} is not trivial, take any nontrivial element g_{3} ∈ Γ_{2}. Let A_{3} ⊂ H_{1} ∩ H_{2} be an axis of it, and H_{3} a wall transversal to A_{3}. Obviously, H_{3} does not contain H_{1} ∩ H_{2}. Let Γ_{3} be the intersection of the stabilizers of H_{1}, H_{2}, and H_{3} in Γ. If Γ_{3} is not trivial, take any nontrivial element g_{4} ∈ Γ_{3}, and so on. The procedure must terminate due to Lemma 3.1.

Thus, we obtain walls H_{1}, H_{2}, ..., H_{k} transversal to axes of some elements of Γ such that the intersection of their stabilizers in Γ is trivial. According to our assumption, the tree T (Γ, H_{i}) is a line for any i. The natural action of Γ on it defines a homomorphism ϕ_{i} of Γ to the infinite dihedral group. The intersection of the kernels of these homomorphisms lies in the intersection of the stabilizers of the walls H_{1}, H_{2}, ..., H_{k} and hence is trivial. Possibly passing to a subgroup of finite index, we may assume that the group ϕ_{i}(Γ) is (infinite) cyclic for any i. Then Γ is abelian.

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Guennadi A. Noskov Institute of Mathematics Pevtsova 13 Omsk 644099 Russia noskov@private.omsk.su Èrnest B. Vinberg Department of Algebra Moscow State University Moscow 119899 Russia vinberg@ebv.pvt.msu.su

Received December 26, 2000 and in final form May 7, 2001