On normal subgroups in automorphism groups

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Aut(AΓ) . In particular, we prove that a finite normal subgroup in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Aut(AΓ) has at most order two and if Γ is not a clique, then any finite normal subgroup in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Aut(AΓ) is trivial. This property has implications for automatic continuity and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> C^{} -algebras: every algebraic epimorphism <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>φ</m:mi> <m:mo lspace="0.278em" rspace="0.278em">:</m:mo> <m:mrow> <m:mi>L</m:mi> <m:mo stretchy="false">↠</m:mo> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> φ L(AΓ) from a locally compact Hausdorff group 𝐿 is continuous if and only if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:math> AΓ is not isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">Z</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> Zⁿ for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> C^{} -algebra of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>A</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Aut(AΓ) . We obtain similar results for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Aut(GΓ) , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:math> GΓ is a graph product of cyclic groups. Moreover, we give a description of the center of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Aut</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>G</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Aut(GΓ) in terms of the defining graph Γ.