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The Boundary and the Virtual Cohomological Dimension of Coxeter Groups

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The Boundary and the Virtual Cohomological Dimension of Coxeter Groups Houston Journal of Mathematics c 2000 University of Houston ­ Volume 26, No. 4, 2000 THE BOUNDARY AND THE VIRTUAL COHOMOLOGICAL DIMENSION OF COXETER GROUPS TETSUYA HOSAKA AND KATSUYA YOKOI Communicated by Charles Hagopian Abstract. This paper consists of three parts: 1) We give some properties about the virtual cohomological dimension of Coxeter groups over principal ideal domains. 2) For a right-angled Coxeter group Γ with vcdR Γ = n, we construct a sequence ΓW0 ΓW1 ΓWn 1 of parabolic subgroups º º ¡ ¡ ¡ º − with vcdR ΓWi = i. 3) We show that a parabolic subgroup of a right-angled Coxeter group is of ¬nite index if and only if their boundaries coincide. 1. Introduction and Preliminaries The purpose of this paper is to study Coxeter groups and their boundaries. Let V be a ¬nite set and m : V V N a function satisfying the following ¢ ! [f1g conditions: (1) m(v; w) = m(w; v) for all v; w V , 2 (2) m(v; v) = 1 for all v V , and 2 (3) m(v; w) 2 for all v = w V . ≥ 6 2 A Coxeter group is a group Γ having the presentation V (vw)m(v;w) = 1 for v; w V ; h j 2 i where if m(v; w) = , then the corresponding relation is omitted, and the pair 1 (Γ;V ) is called a Coxeter system. If m(v; w) = 2 or for all v = w V , then 1 6 2 (Γ;V ) is said to be right-angled. For a Coxeter system (Γ;V ) and a subset W V , º 1991 Mathematics Subject Classi¬cation. 20F65, 20F67, 55M10, 57S30. Key words and phrases. Coxeter group, visual boundary, virtual cohomological dimension, cohomological dimension, dimension. The second author was partially supported by the Ministry of Education, Science, Sports and Culture of Japan under Grant-in-Aid for Encouragement of Young Scientists (No.11740035, 1999-2000). 791 792 T. HOSAKA AND K. YOKOI ΓW is de¬ned as the subgroup of Γ generated by W . The pair (ΓW ;W ) is also a Coxeter system. ΓW is called a parabolic subgroup. For a Coxeter system (Γ;V ), the simplicial complex K(Γ;V ) is de¬ned by the following conditions: (1) the vertex set of K(Γ;V ) is V , and (2) for W = v ; : : : ; v V , v ; : : : ; v spans a k-simplex of K(Γ;V ) if and f 0 kg º f 0 kg only if ΓW is ¬nite. A simplicial complex K is called a ­ag complex if any ¬nite set of vertices, which are pairwise joined by edges, spans a simplex of K. For example, the barycentric subdivision of a simplicial complex is a ­ag complex. For any ¬nite ­ag complex K, there exists a right-angled Coxeter system (Γ;V ) with K(Γ;V ) = K. Namely, let V be the vertex set of K and de¬ne m : V V ¢ ! N by [ f1g 8 > <>1 if v = w, m(v; w) = >2 if v; w spans an edge in K, :> f g otherwise. 1 The associated right-angled Coxeter system (Γ;V ) satis¬es K(Γ;V ) = K. Con- versely, if (Γ;V ) is a right-angled Coxeter system, then K(Γ;V ) is a ¬nite ­ag complex ([7, Corollary 9.4]). For a group Γ and a ring R with identity, the cohomological dimension of Γ over R is de¬ned as cd Γ = sup i Hi (Γ; M) = 0 for some RΓ-module M : R f j 6 g If R = Z then cdZ Γ is simply called the cohomological dimension of Γ, and denoted cd Γ. It is obvious that cd Γ cd Γ for a ring R with identity. It is R ´ known that cd Γ = if Γ is not torsion-free ([5, Corollary VIII 2.5]). A group 1 Γ is said to be virtually torsion-free if Γ has a torsion-free subgroup of ¬nite index. For a virtually torsion-free group Γ the virtual cohomological dimension of Γ over a ring R is de¬ned as cdR Γ0, where Γ0 is a torsion-free subgroup of Γ of ¬nite index, and denoted vcdR Γ. It is a well-de¬ned invariant by Serre's Theorem: if G is a torsion-free group and G0 is a subgroup of ¬nite index, then cdR G0 = cdR G ([5, Theorem VIII 3.1]). If R = Z then vcdZ Γ is simply called the virtual cohomological dimension of Γ, and denoted vcd Γ. It is known that every Coxeter group is virtually torsion-free and the virtual cohomological dimension of each Coxeter group is ¬nite (cf. [6, Corollary 5.2, Proposition 14.1]). BOUNDARY AND DIMENSION OF COXETER GROUPS 793 For a simplicial complex K and a simplex » of K, the closed star St(»; K) of » in K is the union of all simplexes of K having » as a face, and the link Lk(»; K) of » in K is the union of all simplexes of K lying in St(»; K) that are disjoint from ». In [10], Dranishnikov gave the following formula. Theorem 1.1 (Dranishnikov [10]). Let (Γ;V ) be a Coxeter system and R a prin- cipal ideal domain. Then there exists the formula vcd Γ = lcd CK = max lcd K; cd K + 1 ; R R f R R g where K = K(Γ;V ) and CK is the simplicial cone of K. Here, for a ¬nite simplicial complex K and an abelian group G, the local cohomological dimension of K over G is de¬ned as i lcdG K = max» K i H (St(»; K); Lk(»; K); G) = 0 ; 2 f j 6 g and the global cohomological dimension of K over G is cd K = max i H~ i(K; G) = 0 : G f j 6 g i When H~ (K; G) = 0 for each i, then we consider cdG K = 1. We note that i i 1 − H (St(»; K); Lk(»; K); G) is isomorphic to H~ − (Lk(»; K); G). Hence, we have lcdG K = max» K cdG Lk(»; K) + 1 : 2 f g Remark. We recall Dranishnikov's remark in [11]. The de¬nition of the local cohomological dimension in [10] is given by the terminology of the normal star and link. Since Lk(»; K) is homeomorphic to the normal link of » in K, their de¬nitions are equivalent by the formula above. Dranishnikov also proved the following theorem as an application of Theorem 1.1. Theorem 1.2 (Dranishnikov [10]). A Coxeter group Γ has the following proper- ties: (a) vcd Γ vcd Γ for any principal ideal domain R. Q ´ R (b) vcdZp Γ = vcdQ Γ for almost all primes p. (c) There exists a prime p such that vcdZp Γ = vcd Γ. (d) vcd Γ Γ = 2 vcd Γ. ¢ In Section 2, we extend this theorem to one over principal ideal domain coe®- cients. 794 T. HOSAKA AND K. YOKOI Let (Γ;V ) be a Coxeter system and K = K(Γ;V ). Consider the product space Γ CK of Γ with the discrete topology and the underlying space CK ¢ j j j j of the cone of K. De¬ne an equivalence relation on the space as follows: ¸ for (­1; x1); (­2; x2) Γ CK ,(­1; x1) (­2; x2) if and only if x1 = x2 and 1 2 ¢ j j ¸ 1 ­− ­ Γ , where V (x) = v V x St(v; ¬ K) . Here we consider that 1 2 2 V (x1) f 2 j 2 g K is naturally embedded in CK as the base of the cone and ¬1K denotes the j j j j barycentric subdivision of K. The natural left Γ-action on Γ CK is compatible ¢j j with the equivalence relation; hence, it passes to a left action on the quotient space Γ CK = . Denote this quotient space by A(Γ;V ). The space A(Γ;V ) ¢ j j ¸ is contractible and Γ acts cocompactly and properly discontinuously on the space ([6, Theorem 13.5]). We can also give the space A(Γ;V ) a structure of a piecewise Euclidean cell complex with the vertex set Γ v ([7, 9]). Σ(Γ;V ) denotes this piecewise ¢ f 0g x Euclidean cell complex. Refer to [7, De¬nition 2.2] for the de¬nition of a piecewise Euclidean cell complex. In particular, if (Γ;V ) is right-angled, then each cell of Σ(Γ;V ) is a cube, hence, Σ(Γ;V ) is a cubical complex. More precisely, for a right-angled Coxeter system (Γ;V ), we can de¬ne the cubical complex Σ(Γ;V ) by the following conditions: (1) the vertex set of Σ(Γ;V ) is Γ, (2) for ­; ­0 Γ, ­; ­0 spans an edge in Σ(Γ;V ) if and only if the length 1 2 f g lV (­− ­0) = 1, and (3) for ­ Γ and v ; : : : ; v V , the edges ­; ­v ;:::; ­; ­v form a (k +1)- 2 0 k 2 j 0j j kj cube in Σ(Γ;V ) if and only if v ; : : : ; v spans a k-simplex in K(Γ;V ). f 0 kg We note the 1-skeleton of this cell complex is isomorphic to the Cayley graph of Γ with respect to V . For ­ Γ and a k-simplex » = v ; : : : ; v of K(Γ;V ), let 2 j 0 kj C be the (k + 1)-cube in Σ(Γ;V ) formed by ­; ­v ;:::; ­; ­v . Then the ­;» j 0j j kj vertex set of C­;» is ­Γ v ;::: ;v . We note that f 0 kg ¯0 ¯k ­Γ v0;::: ;vk = ­v0 vk ¯i 0; 1 ; i = 0; : : : ; k : f g f ¡¡¡ j 2 f g g For every Coxeter system (Γ;V ), Σ(Γ;V ) is a CAT(0) geodesic space by a piece- wise Euclidean metric (cf.
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