Commensurability classes of hyperbolic Coxeter groups

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...

Geodesics and commensurability classes of arithmetic hyperbolic 3-manifolds

This sharpens [10], where it was shown that the complex length spectrum of M determines its commensurability class. Suppose M ′ is an arithmetic hyperbolic 3-manifold which is not commensurable to M . Theorem 1.1 implies QL(M) 6= QL(M ′), though by Example 2.1 below it is possible that one of QL(M ′) or QL(M) contains the other. By the length formulas recalled in §2.1 and §2.2, each element of ...

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.