### 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,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 m bounded from above.

Original language | English |
---|---|

Pages (from-to) | 701-736 |

Number of pages | 36 |

Journal | Groups, Geometry, and Dynamics |

Volume | 6 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 |

### Fingerprint

### 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

*Groups, Geometry, and Dynamics*,

*6*(4), 701-736. https://doi.org/10.4171/GGD/171

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

Research output: Contribution to journal › Article

*Groups, Geometry, and Dynamics*, vol. 6, no. 4, pp. 701-736. https://doi.org/10.4171/GGD/171

}

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 -