Graphs and complexes of lattices

Abstract

We study lattices acting on $\mathrm{CAT}(0)$ spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices and a complex of lattices giving graph and complex of group splittings of $\mathrm{CAT}(0)$ lattices. Using this framework we characterise irreducible uniform $(\mathrm{Isom}({\mathbb{E}}^n)\times T)$-lattices by $C^{\ast }$-simplicity and give a necessary condition for lattices in products with a Euclidean factor to be biautomatic. We also construct non-residually finite uniform lattices acting on arbitrary products of right-angled buildings and non-biautomatic lattices acting on the product of ${\mathbb{E}}^n$ and a right-angled building.

v5 Version accepted for publication in Algebraic and Geometric Topology; v4. Major rewrite, paper split at request of referee (see bibliographical note in introduction for details). 38 pages