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

References in corpus (9)

• Lattices acting on right-angled buildings
• Mapping tori of free group automorphisms, and the Bieri-Neumann-Strebel invariant of graphs of groups
• Bounding the homological finiteness length
• On the fundamental group of certain polyhedral products
• $L^2$-Betti numbers of totally disconnected groups and their approximation by Betti numbers of lattices
• Measure equivalence classification of transvection-free right-angled Artin groups
• New simple lattices in products of trees and their projections
• Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes
• Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups"

Cited by in corpus (3)

• Lattices in a product of trees, hierarchically hyperbolic groups, and virtual torsion-freeness
• Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity
• A note on the rational homological dimension of lattices in positive characteristic