We show that every abstract homomorphism ϕ from a locally compact group L to a graph product G Γ , endowed with the discrete topology, is either continuous or ϕ(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 ϕ : L → G is either continuous, or ϕ(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.

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We study algebraic and geometric properties of the automorphism group of universal Coxeter group of rank n ≥ 4, Aut(Wn). In particular, we prove that whenever Aut(Wn) acts by isometries on complete d-dimensional CAT(0) space with d < ⌊ n 2 ⌋, then it must fix a point. We also prove that Aut(Wn) does not have Kazhdan's property (T) and is not amenable. Further, strong restrictions are obtained on homomorphisms of Aut(Wn) to groups that do not contain a copy of Sym(n).

Given an Artin group A Γ , a common strategy in the study of A Γ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that A Γ has a specific property if and only if all "small" parabolic subgroups of A Γ have this property. Since "small" parabolic subgroups are the puzzle pieces of A Γ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of A Γ 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 A Γ to fixed point properties and to automatic continuity of A Γ using Bass-Serre theory and a generalization of the Deligne complex.

We deduce from Sageev's 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 partially prove a conjecture by Marquis concerning actions on buildings and 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.