Research news
The preprint "On normal subgroups in automorphism groups" by Philip Möller and Olga Varghese has been recently uploaded to the arXiv!
Link to the paper: arXiv:2208.05677
Abstract:
We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut(AΓ). In particular, we prove that a finite normal subgroup in Aut(AΓ) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut(AΓ) is trivial. This property has implications to automatic continuity and to C∗-algebras: every algebraic epimorphism φ:L↠Aut(AΓ) from a locally compact Hausdorff group L is continuous if and only if AΓ 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(AΓ). We obtain similar results for Aut(GΓ) where GΓ is a graph product of cyclic groups. Moreover, we give a description of the center of Aut(GΓ) in terms of the defining graph Γ.