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 Aut ⁢ (A Γ) Aut (A_). In particular, we prove that a finite normal subgroup in Aut ⁢ (A Γ) Aut (A_) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ⁢ (A Γ) Aut (A_) is trivial. This property has implications for automatic continuity and C ∗ C^ -algebras: every algebraic epimorphism φ: L ↠ Aut ⁢ (A Γ) L (A_) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_ is not isomorphic to Z n Z^n for any n ≥ 1 n 1. Furthermore, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C ∗ C^ -algebra of Aut ⁢ (A Γ) Aut (A_). We obtain similar results for Aut ⁢ (G Γ) Aut (G_), where G Γ G_ is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ⁢ (G Γ) Aut (G_) in terms of the defining graph Γ.