Graphs and complexes of lattices

Abstract

We study lattices acting on $\textrm{CAT}(0)$ 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 $\textrm{CAT}(0)$ lattices. Using this framework we characterise irreducible uniform $(\textrm{Isom}(\mathbb{E}^n)\times T)$-lattices by $C^\ast$-simplicity and the failure of virtual fibring and biautomaticity. We 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. We investigate the residual finiteness, $L^2$-cohomology, and $C^\ast$-simplicity of $\textrm{CAT}(0)$ lattices more generally. Along the way we prove that many right angled Artin groups with rank $2$ centre are not quasi-isometrically rigid.

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ORCID for Sam Hughes: orcid.org/0000-0002-9992-4443

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