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

2012/10/22 by Hiroyasu Izeki, Takefumi Kondo, Izeki, Hiroyasu +3
Mathematics · #20F65 (Primary) 58E20 #20P05 (Secondary) #Advanced Operator Algebra Research #Advanced Topology and Set Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.1210.5829

openalex publication_date 2012/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that a random group of the graph model associated with a sequence of expanders has fixed-point property for a certain class of CAT(0) spaces. We use Gromov's criterion for 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, to 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 fixed-point property for all of these Euclidean buildings with m bounded from above.

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