On normal subgroups in automorphism groups

TL;DR

This paper investigates the structure of normal subgroups in automorphism groups of right-angled Artin groups, revealing properties about finite normal subgroups, implications for continuity of homomorphisms, and the structure of the automorphism group's center.

Contribution

It characterizes finite normal subgroups in automorphism groups of right-angled Artin groups and explores their implications for algebraic and topological properties.

Findings

Finite normal subgroups have order at most two.

If the defining graph is not a clique, finite normal subgroups are trivial.

Invertible elements are dense in the reduced group C*-algebra under certain conditions.

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 $\Gamma$.