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

Hiroyasu Izeki, Takefumi Kondo, Shin Nayatani

Research output: Contribution to journalArticle

6 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)701-736
Number of pages36
JournalGroups, Geometry, and Dynamics
Volume6
Issue number4
DOIs
Publication statusPublished - 2012

Fingerprint

Fixed Point Property
CAT(0) Spaces
Euclidean
Energy
Equivariant Map
Tangent Cone
Geometric Invariants
Expander
Finitely Generated Group
Graph Model
Deduce
Estimate
Buildings

Keywords

  • CAT(0) space
  • Energy of map
  • Euclidean building
  • Expander
  • Finitely generated group
  • Fixed-point property
  • Random group
  • Wang invariant

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

N-step energy of maps and the fixed-point property of random groups. / Izeki, Hiroyasu; Kondo, Takefumi; Nayatani, Shin.

In: Groups, Geometry, and Dynamics, Vol. 6, No. 4, 2012, p. 701-736.

Research output: Contribution to journalArticle

Izeki, Hiroyasu ; Kondo, Takefumi ; Nayatani, Shin. / N-step energy of maps and the fixed-point property of random groups. In: Groups, Geometry, and Dynamics. 2012 ; Vol. 6, No. 4. pp. 701-736.
@article{7c86ddcef4244e5fabaf68b2f487c5bd,
title = "N-step energy of maps and the fixed-point property of random groups",
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 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.",
keywords = "CAT(0) space, Energy of map, Euclidean building, Expander, Finitely generated group, Fixed-point property, Random group, Wang invariant",
author = "Hiroyasu Izeki and Takefumi Kondo and Shin Nayatani",
year = "2012",
doi = "10.4171/GGD/171",
language = "English",
volume = "6",
pages = "701--736",
journal = "Groups, Geometry, and Dynamics",
issn = "1661-7207",
publisher = "European Mathematical Society Publishing House",
number = "4",

}

TY - JOUR

T1 - N-step energy of maps and the fixed-point property of random groups

AU - Izeki, Hiroyasu

AU - Kondo, Takefumi

AU - Nayatani, Shin

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - CAT(0) space

KW - Energy of map

KW - Euclidean building

KW - Expander

KW - Finitely generated group

KW - Fixed-point property

KW - Random group

KW - Wang invariant

UR - http://www.scopus.com/inward/record.url?scp=84872401107&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872401107&partnerID=8YFLogxK

U2 - 10.4171/GGD/171

DO - 10.4171/GGD/171

M3 - Article

VL - 6

SP - 701

EP - 736

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

SN - 1661-7207

IS - 4

ER -