Kazhdan–Lusztig cells in planar hyperbolic Coxeter groups and automata

Kazhdan-Lusztig cells in planar hyperbolic Coxeter groups and automata

Let C be a oneor two-sided Kazhdan–Lusztig cell in a Coxeter group (W,S), and let Reduced(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is regular. In this paper we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Ca...

Coxeter Groups and Hyperbolic Manifolds

In the last quarter of a century, 3-manifold topology has been revolutionized by Thurston and his school. This has generated a huge literature on hyperbolic 3-manifolds, building on the classical body of knowledge already existing in 2-dimensions. Balanced against this is a relative paucity of techniques and examples of hyperbolic n-manifolds for n ≥ 4. Recent work of Ratcliffe and Tschantz has...

On Subgroup Separability in Hyperbolic Coxeter Groups

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 s...

Mathematische Annalen Coxeter groups and hyperbolic manifolds

The theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic manifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4, 5 and 6-dimensions, are given, each of very small volume, and in one case of smallest possible volume.