Automatic continuity for groups whose torsion subgroups are small

We prove that a group homomorphism $varphicolon Lto G$ from a locally compact Hausdorff group $L$ into a discrete group $G$ either is continuous, or there exists a normal open subgroup $Nsubseteq L$ such that $varphi(N)$ is a torsion group provided that $G$ does not include $mathbb{Q}$ or the $p$-adic integers $mathbb{Z}_p$ or the Prufer $p$-group $mathbb{Z}(p^infty)$ for any prime $p$ as a subgroup, and if the torsion subgroups of $G$ are small in the sense that any torsion subgroup of $G$ is artinian. In particular, if $varphi$ is surjective and $G$ additionaly does not have non-trivial normal torsion subgroups, then $varphi$ is continuous. As an application we obtain results concerning the continuity of group homomorphisms from locally compact Hausdorff groups to many groups from geometric group theory, in particular to automorphism groups of right-angled Artin groups and to Helly groups.

Abstract group actions of locally compact groups on CAT(0) spaces

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.

Automorphism groups of Coxeter groups do not have Kazhdans property (T)

We show that for a large class $mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Gamma$ in $mathcal{W}$, the automorphism group ${rm Aut}(W_Gamma)$ virtually surjects onto $W_Gamma$. In particular, the group ${rm Aut}(G_Gamma)$ is virtually indicable and therefore does not satisfy Kazhdans property (T). Moreover, if $W_Gamma$ is not virtually abelian, then the group ${rm Aut}(W_Gamma)$ is large.

On number of ends of graph products of groups

Given a finite simplicial graph $Gamma=(V,E)$ with a vertex-labelling $varphi:Vrightarrowleft{text{non-trivial finitely generated groups}right}$, the graph product $G_Gamma$ is the free product of the vertex groups $varphi(v)$ with added relations that imply elements of adjacent vertex groups commute. For a quasi-isometric invariant $mathcal{P}$, we are interested in understanding under which combinatorial conditions on the graph $Gamma$ the graph product $G_Gamma$ has property $mathcal{P}$. In this article our emphasis is on number of ends of a graph product $G_Gamma$. In particular, we obtain a complete characterization of number of ends of a graph product of finitely generated groups.

On hyperbolicity and virtual freeness of automorphism groups

We show that word hyperbolicity of automorphism groups of graph products $G_Gamma$ and of Coxeter groups $W_Gamma$ depends strongly on the shape of the defining graph $Gamma$. We also characterized those $Aut(G_Gamma)$ and $Aut(W_Gamma)$ in terms of $Gamma$ that are virtually free.

Abstract homomorphisms from locally compact groups to discrete groups

We show that every abstract homomorphism $varphi$ from a locally compact group $L$ to a graph product $G_Gamma$, endowed with the discrete topology, is either continuous or $varphi(L)$ lies in a small parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not small is discrete. This extends earlier work by Morris-Nickolas. We also show the following. If $L$ is a locally compact group and if $G$ is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism $varphi:Lto G$ is either continuous, or $varphi(L)$ is contained in the normalizer of a finite nontrivial subgroup of $G$. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.

On property (T) for automorphism groups of graph products

We show that the automorphism group of a graph product of finite groups $Aut(G_Gamma)$ has Kazhdans property (T) if and only if $Gamma$ is a complete graph.

A note on locally elliptic actions on cube complexes

We deduce from Sageevs results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji.

Planarity of Cayley graphs of graph products of groups

We obtain a complete classification of graph products of finite abelian groups whose Cayley graphs with respect to the standard presentations are planar.

On coherence of graph products and Coxeter groups

We study coherence of graph products and Coxeter groups and obtain many results in this direction.