### 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 |
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Pages (from-to) | 701-736 |

Number of pages | 36 |

Journal | Groups, Geometry, and Dynamics |

Volume | 6 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2012 Dec 1 |

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology
- Discrete Mathematics and Combinatorics

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## Cite this

Izeki, H., Kondo, T., & Nayatani, S. (2012). N-step energy of maps and the fixed-point property of random groups.

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