On normal subgroups in automorphism groups

2022/08/11 by Möller, Philip, Varghese, Olga
#FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Primary: 20E36 #Secondary: 20E06

paper · doi:10.48550/arxiv.2208.05677

Abstract

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group \rm Aut(A_Γ). In particular, we prove that a finite normal subgroup in \rm Aut(A_Γ) has at most order two and if Γ is not a clique, then any finite normal subgroup in \rm Aut(A_Γ) is trivial. This property has implications to automatic continuity and to C^∗-algebras: every algebraic epimorphism φ\colon L\twoheadrightarrow\rm Aut(A_Γ) from a locally compact Hausdorff group L is continuous if and only if A_Γ is not isomorphic to ℤⁿ for any n≥ 1. Further, 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^∗-algebra of Aut(A_Γ). We obtain similar results for \rm Aut(G_Γ) where G_Γ is a graph product of cyclic groups. Moreover, we give a description of the center of Aut(G_Γ) in terms of the defining graph Γ.

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