Geodesic
$N$-step energy of maps and the fixed-point property of random groups
Hiroyasu Izeki1 ; Takefumi Kondo2 ; Shin Nayatani3
1Keio University, Yokohama, Japan
2Kobe University, Japan
3Nagoya University, Japan
Groups, geometry, and dynamics, Tome 6 (2012) no. 4, pp. 701-736

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DOI

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,Qr​), 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.
DOI : 10.4171/ggd/171
Classification : 20-XX, 58-XX, 00-XX
Mots-clés : Finitely generated group, random group, CAT(0) space, fixed-point property, energy of map, Wang invariant, expander, Euclidean building

Hiroyasu Izeki  1   ; Takefumi Kondo  2   ; Shin Nayatani  3

1 Keio University, Yokohama, Japan
2 Kobe University, Japan
3 Nagoya University, Japan 
Hiroyasu Izeki; Takefumi Kondo; Shin Nayatani. $N$-step energy of maps and the fixed-point property  of random groups. Groups, geometry, and dynamics, Tome 6 (2012) no. 4, pp. 701-736. doi: 10.4171/ggd/171
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     title = {$N$-step energy of maps and the fixed-point property  of random groups},
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     doi = {10.4171/ggd/171},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/171/}
}
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