Abstract group actions of locally compact groups on CAT(0) spaces
Abstract: We study abstract group actions of locally compact Hausdorff groups on CAT.0/ spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion-free CAT.0/ group is continuous.
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Cited by 2 publications
References 28 publications
On parabolic subgroups of Artin groups
Given an Artin group AΓ, a common strategy in the study of AΓ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e., showing that AΓ has a specific property if and only if all “small” parabolic subgroups of AΓ have this property. Since “small” parabolic subgroups are the building blocks of AΓ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of AΓ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of AΓ to fixed point properties and to automatic continuity of AΓ using Bass–Serre theory and a generalization of the Deligne complex.
On parabolic subgroups of Artin groups
Given an Artin group AΓ, a common strategy in the study of AΓ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e., showing that AΓ has a specific property if and only if all “small” parabolic subgroups of AΓ have this property. Since “small” parabolic subgroups are the building blocks of AΓ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of AΓ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of AΓ to fixed point properties and to automatic continuity of AΓ using Bass–Serre theory and a generalization of the Deligne complex.
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 Aut ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . In particular, we prove that a finite normal subgroup in Aut ( A Γ ) \mathrm{Aut}(A_{\Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ( A Γ ) \mathrm{Aut}(A_{\Gamma}) is trivial. This property has implications for automatic continuity and C ∗ C^{\ast} -algebras: every algebraic epimorphism φ : L ↠ Aut ( A Γ ) \varphi\colon L\twoheadrightarrow\mathrm{Aut}(A_{\Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{\Gamma} is not isomorphic to Z n \mathbb{Z}^{n} for any n ≥ 1 n\geq 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^{\ast} -algebra of Aut ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . We obtain similar results for Aut ( G Γ ) \mathrm{Aut}(G_{\Gamma}) , where G Γ G_{\Gamma} is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ( G Γ ) \mathrm{Aut}(G_{\Gamma}) in terms of the defining graph Γ.
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