On normal subgroups in automorphism groups (Q6661333)

A group \(G\) is called full-sized if it contains a non-abelian free group \(F_{2} \hookrightarrow G\) (such terminology was introduced in [\textit{M. R. Bridson} et al., Ann. Math. (2) 192, No. 3, 679--719 (2020; Zbl 1455.57028)]).\N\NIn the paper under review, the authors are interested in the structure of non-full sized normal subgroups in the automorphism group of a group that is defined by a graph. Let \(\Gamma=(V,E)\) be a finite simplicial graph and let \(A_{\Gamma}\) be the right-angled Artin group defined by \(\Gamma\). A first result obtained by the authors is Proposition A: Let \(\mathrm{Aut}(A)\) be the automorphism group of a centerless rightangled Artin group \(A_{\Gamma}\) and let \(N \trianglelefteq \mathrm{Aut}(A)\) be a normal subgroup. If \(N\) is nontrivial, then \(N\) is full-sized.\N\NThe main result of the paper is Theorem B, whose statement is too long to be included here. Two important consequences of this theorem are the following:\N\NCorollary C: The automorphism group of a right-angled Artin group \(\mathrm{Aut}(A_{\Gamma})\) has non-trivial finite normal subgroups if and only if \(\Gamma\) is a clique.\N\NCorollary D: The automorphism group of a right-angled Artin group \(\mathrm{Aut}(A_{\Gamma})\) is centerless if and only if \(\Gamma\) is a clique.\N\NThese results have implications for automatic continuity and \(C^{\ast}\)-algebras: every algebraic epimorphism \(\varphi: L \twoheadrightarrow \mathrm{Aut}(A_{\Gamma})\) from a locally compact Hausdorff group \(L\) is continuous if and only if \(A_{\Gamma}\) is not isomorphic to \(\mathbb{Z}_{n}\) for any \(n \geq 1\). Furthermore, 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 \(\mathrm{Aut}(A_{\Gamma})\).