# Groups, Geometry, and Dynamics

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**Volume 6, Issue 4, 2012, pp. 701–736**

**DOI: 10.4171/GGD/171**

Published online: 2012-11-10

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

^{[1]}, Takefumi Kondo

^{[2]}and Shin Nayatani

^{[3]}(1) Keio University, Yokohama, Japan

(2) Kobe University, Japan

(3) Nagoya University, Japan

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 Gromovâ€™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.

*Keywords: *Finitely generated group, random group, CAT(0) space, fixed-point property, energy of map, Wang invariant, expander, Euclidean building

Izeki Hiroyasu, Kondo Takefumi, Nayatani Shin: `N`-step energy of maps and the fixed-point property of random groups. *Groups Geom. Dyn.* 6 (2012), 701-736. doi: 10.4171/GGD/171