Coxeter groups and random groups

TL;DR

This paper explores the relationship between Coxeter groups and random groups, showing that random groups at certain densities almost surely contain specific quasiconvex subgroups related to hyperbolic reflection groups.

Contribution

It establishes a connection between random groups and superideal reflection groups, demonstrating the presence of particular quasiconvex subgroups in random groups at various densities.

Findings

Random groups contain quasiconvex subgroups related to hyperbolic reflection groups.

Superideal reflection groups exist in all dimensions as convex co-compact groups.

With high probability, random groups at density less than 1/2 include these subgroups.

Abstract

For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgroups commensurable with some member of the family, with overwhelming probability.