JournalsggdVol. 6, No. 4pp. 701–736

<var>N</var>-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
<var>N</var>-step energy of maps and the fixed-point property  of random groups cover
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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 nn-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,Qrm,\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 mm bounded from above.

Cite this article

Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani, <var>N</var>-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