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

### Hiroyasu Izeki

Keio University, Yokohama, Japan### Takefumi Kondo

Kobe University, Japan### Shin Nayatani

Nagoya University, Japan

## 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($m,Q_{r}$), 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.

## Cite this article

Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani, $N$-step energy of maps and the fixed-point property of random groups. Groups Geom. Dyn. 6 (2012), no. 4, pp. 701–736

DOI 10.4171/GGD/171