Commensurability and separability of quasiconvex subgroups

TL;DR

This paper investigates the conditions under which uniform lattices in various hyperbolic and CAT(0) spaces are commensurable, emphasizing the roles of quasiconvex subgroup separability and geometric properties.

Contribution

It establishes new criteria linking quasiconvex subgroup separability to the commensurability of lattices in hyperbolic and CAT(0) spaces, extending previous results.

Findings

Uniform lattices in regular right-angled Fuchsian buildings with polygons of at least six edges are commensurable.

In hyperbolic right-angled buildings associated with graph products, lattices are commensurable if all quasiconvex subgroups are separable.

Extensions of uniform lattices in CAT(0) square complexes by finite groups are virtually trivial under quasiconvex subgroup separability.

Abstract

We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with the graph product provided all of its quasiconvex subgroups are separable. We obtain a similar result for uniform lattices of the Davis complex of Gromov-hyperbolic two-dimensional Coxeter groups. We also prove that every extension of a uniform lattice of a CAT(0) square complex by a finite group is virtually trivial, provided each quasiconvex subgroup of the lattice is separable.