Summary: On Subgroup Separability in Hyperbolic
D. D. LONG1c
and A. W. REID2cc
Department of Mathematics, UCSB, Santa Barbara, CA 93106, U.S.A.
Department of Mathematics, University of Texas at Austin, Austin, TX 78712, U.S.A.
(Received: 25 May 2000; in ˘nal form: 20 February 2001)
Abstract. We prove that certain hyperbolic Coxeter groups are separable on their geometrically
Mathematics Subject Classi˘cation (2000). 20H10.
Key words. hyperbolic Coxeter group, subgroup separability.
Recall that a subgroup H of a group G is separable in G if, given any g P G n H, there
exists a subgroup K ` G of ˘nite index with H ` K and g aP K. 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 , and  for example). In that context, and in the context of negatively curved