On Subgroup Separability in Hyperbolic Coxeter Groups

We prove that certain hyperbolic Coxeter groups are separable on their geometrically ¢nite subgroups. Mathematics Subject Classi¢cation (2000). 20H10. Key words. hyperbolic Coxeter group, subgroup separability. 1. Introduction Recall that a subgroupH of a group G is separable in G if, given any g 2 G nH, there exists a subgroup K < G of ¢nite index with H < K and g = 2K . G is called subgroup separable (or LERF) if G is H-subgroup separable for all ¢nitely generated H < G. This powerful property has attracted a good deal of attention in the last few years, largely motivated by questions which arise in low dimensional topology (see [1], and [15] for example). In that context, and in the context of negatively curved groups it makes most sense to restrict to subgroups which are geometrically ¢nite (or quasiconvex in the negatively curved case) and to this end, the notion of GFERFwas introduced in [1] and [12]. Since the property of being geometrically ¢nite is preserved by passage to suband super-group of ¢nite index, as in the case of subgroup separability (see [15]), it follows that GFERF is a commensurability invariant. This paper studies GFERF in the context of arithmetic groups and certain Coxeter groups which we discuss further below. However, this paper should really be viewed as a broad generalization of the geometric and algebraic methods used in [1], in which the theory of quadratic forms was used to help control properties of discrete groups. Throughout this paper we will use the term hyperbolic simplex group to refer to those Coxeter groups which arise as groups generated by re£ections in the faces of a non-compact geodesic hyperbolic simplex of ¢nite volume. Thus, these Workwas partially supported by the NSF. Workwas partially supported by the NSF,The Alfred P. Sloan Foundation and a grant from the Texas Advanced Research Program. Geometriae Dedicata 87: 245^260, 2001. 245 # 2001 Kluwer Academic Publishers. Printed in the Netherlands. hyperbolic simplex groups are ¢nite co-volume but non-cocompact discrete subgroups of Isom…Hn† for some n. The general de¢nition of a Coxeter group is given in Section 2. As is well-known, there are very few hyperbolic simplex groups, and these are completely classi¢ed; they exist only in dimensions 3W nW 9 (see [7], pp. 142^144). A list of the Coxeter diagrams of such groups in dimensions 4W nW 8 is given in the Appendix. In dimension 3, there are 23 such hyperbolic simplex groups, and all but 6 are arithmetic. The arithmetic ones are commensurable with either of the Bianchi groups PSL…2;Z‰iŠ† or PSL…2;Z‰oŠ† where o is a cube root of unity (see [8] or [14], for example). Thus by [1] these arithmetic hyperbolic simplex groups are GFERF. In dimensions X 4 all the commensurability classes with one exception (in dimension 5) are arithmetic (see below and [17]). This paper shows that the commensurability classes of arithmetic hyperbolic simplex groups in dimensions W 8 are GFERF. THEOREM 1.1. Let G be an arithmetic ¢nite volume non-cocompact hyperbolic simplex group of dimension W 8. Then G is GFERF. The main calculation is summed up by the following resultösee Section 2 for notation and de¢nitions. THEOREM 1.2. Let G be an arithmetic hyperbolic simplex group of dimension 4W nW 9 and F …G† the rational form constructed in Section 5. Suppose that the determinant of the form F …G† is ÿk. Then F …G† is equivalent over Q to h1; . . . ; 1;ÿki. In particular, either F …G† (in the case k ˆ 1) or F …G† hki (otherwise) is equivalent over Q to the standard form h1; . . . ; 1;ÿ1i. & The method of proof for Theorem 1.1 is closely related to that of [1]; as in that paper we observe that there are hyperbolic simplex groups in dimensions 6, 7, 8 which are commensurable with a group generated by re£ections in an all right polyhedron. Theorem 3.1 of [1] now shows that this latter group is GFERF, so that the hyperbolic simplex group in the appropriate dimension is GFERF. It is then shown from certain arithmetic considerations that every hyperbolic arithmetic simplex group, is already commensurable with one of these groups, and so is GFERF, or can be embedded into one of these groups, hence inherits the GFERF property. There are arithmetic hyperbolic simplex groups in dimension 9, but we do not know if these are commensurable with all right re£ection groups, and hence our results concerning GFERF are limited to dimensions W 8. As will be apparent, a central role in our methods is played by the groups SO0… fn;Z†; where fn denotes the form h1; . . . ; 1;ÿ1i of signature …n; 1†. Indeed, it appears that the family of groups SO0…fn;Z† form a potentially important `universal family' of groups in the following sense. 246 D. D. LONG AND A.W. REID