On normal subgroups in automorphism groups

Abstract

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group ${\rm Aut}(A_\Gamma)$. In particular, we prove that a finite normal subgroup in ${\rm Aut}(A_\Gamma)$ has at most order two and if $\Gamma$ is not a clique, then any finite normal subgroup in ${\rm Aut}(A_\Gamma)$ is trivial. This property has implications to automatic continuity and to $C^\ast$-algebras: every algebraic epimorphism $\varphi\colon L\twoheadrightarrow{\rm Aut}(A_\Gamma)$ from a locally compact Hausdorff group $L$ is continuous if and only if $A_\Gamma$ is not isomorphic to $\mathbb{Z}^n$ for any $n\geq 1$. Further, if $\Gamma$ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group $C^\ast$-algebra of Aut$(A_\Gamma)$. We obtain similar results for ${\rm Aut}(G_\Gamma)$ where $G_\Gamma$ is a graph product of cyclic groups. Moreover, we give a description of the center of Aut$(G_\Gamma)$ in terms of the defining graph $\Gamma$.