Fixed point property for a CAT(0) space which admits a proper cocompact group action

Abstract: We prove that if a geodesically complete $\mathrm{CAT}(0)$ space $X$ admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of $X$ is less than $1$. Let $G$ be a finite connected graph, $\mu_1 (G)$ be the linear spectral gap of $G$, and $\lambda_1 (G,X)$ be the nonlinear spectral gap of $G$ with respect to such a $\mathrm{CAT}(0)$ space $X$. Then, the result implies that the ratio $\lambda_1 (G,X) / \mu_1 (G)$ is bounded from below by a positive constant which is independent of the graph $G$. It follows that any isometric action of a random group of the graph model on such $X$ has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.