Lattice envelopes of right-angled Artin groups

Abstract

Let $\Gamma$ be a finite simplicial graph with at least two vertices, and let $G(\Gamma)$ be the associated right-angled Artin group. We describe a locally compact group $\mathcal U$ containing $G(\Gamma)$ as a cocompact lattice. If $\Gamma$ is not a join (i.e. the complement graph is connected), the group $\mathcal U$ is non-discrete, almost simple, but not virtually simple: it has a smallest normal subgroup $\mathcal U^+$ which is an open simple subgroup, and the quotient $\mathcal U/\mathcal U^+$ is isomorphic to the right-angled Coxeter group $W(\Gamma)$. Under suitable assumptions on $\Gamma$, we rely on work by Bader-Furman-Sauer and Huang-Kleiner to show that $\mathcal U \rtimes \mathrm{Aut}(\Gamma)$ is the universal lattice envelope of $G(\Gamma)$: for every lattice envelope $H$ of $G(\Gamma)$, there is a continuous proper homomorphism $H \to \mathcal U \rtimes \mathrm{Aut}(\Gamma)$. In particular, no lattice envelope of $G(\Gamma)$ is virtually simple. We also show that no locally compact group quasi-isometric to $G(\Gamma)$ is virtually simple. This contrasts with the case of free groups. The group $\mathcal U$ is a universal automorphism group of the Davis building of $G(\Gamma)$, with prescribed local actions. As an application, we describe the algebraic structure of the full automorphism group of the Cayley graph of $G(\Gamma)$ with respect to its standard generating set.