Rigidity of matrix group actions on CAT(0) spaces with possible parabolic isometries and uniquely arcwise connected spaces

Abstract

It is well-known that $\mathrm{SL}_{n}(\mathbf{Q}_{p})$ acts without fixed points on an $(n-1)$-dimensional $\mathrm{CAT}(0)$ space (the affine building). We prove that $n-1$ is the smallest dimension of $\mathrm{CAT}(0)$ spaces on which matrix groups act without fixed points. Explicitly, let $R$ be an associative ring with identity and $E_{n}^{\prime }(R)$ the extended elementary subgroup. Any isometric action of $E_{n}^{\prime }(R)$ on a complete $\mathrm{CAT(0)}$ space $X^{d}$ of dimension $d<n-1$ has a fixed point. Similar results are discussed for automorphism groups of free groups. Furthermore, we prove that any action of $\mathrm{Aut}(F_{n}),n\geq 3,$ on a uniquely arcwise connected space by homeomorphisms has a fixed point.