#### Volume 6, issue 2 (2006)

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Commensurability and separability of quasiconvex subgroups

### Frédéric Haglund

Algebraic & Geometric Topology 6 (2006) 949–1024
 arXiv: 0904.2698
##### 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 $CAT\left(0\right)$ square complex by a finite group is virtually trivial, provided each quasiconvex subgroup of the lattice is separable.

##### Keywords
graph products, Coxeter groups, commensurability, separability, quasiconvex subgroups, right-angled buildings, Davis' complexes, finite extensions
##### Mathematical Subject Classification 2000
Primary: 20F55, 20F67, 20F65
Secondary: 20E26, 51E24, 20E22, 20J06