Commensurability and separability of quasiconvex subgroups

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. 20F55, 20F67, 20F65; 20E26, 51E24, 20E22, 20J06