Two topics in geometric group theory

Abstract

Niblo and Reeves [NR2] constructed a cubing for each Coxeter group using the hyperplanes of the Coxeter complex. In Part 1 Coxeter groups and cubings the natural action of the Coxeter group on this cubing is investigated. In particular the cocompactness or not of this action is studied. Using the geometry of the Moussong complex (another complex for a Coxeter group introduced by Gabor Moussong in [Mou]) it is shown that hyperbolic and right-angled Coxeter groups act cocompactly and Euclidean Coxeter groups act non-cocompactly and that the action is non-cocompact if and only if there exists an infinite family of non-conjugate isomorphic triangle subgroups.

In Part II Engulfing and subgroup separability for word-hyperbolic groups theorems of Darren Long [L] concerning fundamental groups of closed hyperbolic manifolds are generalised to word-hyperbolic groups. The main result is that if a torsion-free word-hyperbolic group has a certain engulfing property then every quasiconvex subgroup is contained as a finite index subgroup in a separable subgroup.

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