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.
Categories: math.GT math.DS