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

Abstract

We prove that if a geodesically complete 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, μ₁(G) be the linear spectral gap of G, and λ₁(G,X) be the nonlinear spectral gap of G with respect to such a CAT(0) space X. Then, the result implies that the ratio λ₁(G,X)/μ₁(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.