Commensurability and separability of quasiconvex subgroups

Haglund, Frédéric

doi:10.2140/agt.2006.6.949

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ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

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Commensurability and separability of quasiconvex subgroups

Frédéric Haglund

Algebraic & Geometric Topology 6 (2006) 949–1024

DOI: 10.2140/agt.2006.6.949

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 $CAT (0)$ 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

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Publication

Received: 25 July 2005

Accepted: 23 November 2005

Published: 9 August 2006

Authors

Frédéric Haglund
Laboratoire de Mathématiques Université de Paris XI (Paris-Sud) 91405 Orsay France

Volume 6, issue 2 (2006)