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 -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(), and deduce from the general result above that the same random group has the fixed-point property for all of these Euclidean buildings with bounded from above.
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Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani, -step energy of maps and the fixed-point property of random groups. Groups Geom. Dyn. 6 (2012), no. 4, pp. 701–736DOI 10.4171/GGD/171