$N$-step energy of maps and the fixed-point property of random groups

Abstract

We prove that a random group of the graph model associated with a sequence of expanders has the fixed-point property for a certain class of CAT(0) spaces. We use Gromov’s criterion for the fixed-point property in terms of the growth of $n$-step energy of equivariant maps from a finitely generated group into a CAT(0) space, for which we give a detailed proof. We estimate a relevant geometric invariant of the tangent cones of the Euclidean buildings associated with the groups PGL($m,{\mathbb{Q}}_r$), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with $m$ bounded from above.