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

It is well-known that ${\mathrm{S}\mathrm{L}}_n({\mathbf{Q}}_p)$ acts without fixed points on an $(n-1)$-dimensional $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ space (the affine building). We prove that $n-1$ is the smallest dimension of $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ spaces on which matrix groups act without fixed points. Explicitly, let $R$ be an associative ring with identity and $E_n^{\mathrm{\prime }}(R)$ the extended elementary subgroup. Any isometric action of $E_n^{\mathrm{\prime }}(R)$ on a complete $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ 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 $\mathrm{A}\mathrm{u}\mathrm{t}(F_n),n\ge 3,$ on a uniquely arcwise connected space by homeomorphisms has a fixed point.