Fixed points for actions of Aut(Fn) on CAT(0) spaces

For n greater or equal 4 we discuss questions concerning global fixed points for isometric actions of Aut(Fn), the automorphism group of a free group of rank n, on complete CAT(0) spaces. We prove that whenever Aut(Fn) acts by isometries on complete d-dimensional CAT(0) space with d is less than 2 times the integer function of n over 4 and minus 1, then it must fix a point. This property has implications for irreducible representations of Aut(Fn), which are also presented here. For SAut(Fn), the unique subgroup of index two in Aut(Fn), we obtain similar results

Conjugating automorphisms of graph products: Kazhdan's property (T) and SQ-universality

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan's property (T) and when it satisfies some vastness properties including SQ-universality.Comment: 8 pages, comments are welcome

Automorphism groups of Coxeter groups do not have Kazhdan's 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 Kazhdan's property (T). Moreover, if $W_\Gamma$ is not virtually abelian, then the group ${\rm Aut}(W_\Gamma)$ is large