## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 39, Number 1 (2016), 129-153.

### Fixed point property for a CAT(0) space which admits a proper cocompact group action

#### Abstract

We prove that if a geodesically complete CAT(0) space *X* admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of *X* is less than 1. Let *G* be a finite connected graph, μ_{1}(*G*) be the linear spectral gap of *G*, and λ_{1}(*G,X*) be the nonlinear spectral gap of *G* with respect to such a CAT(0) space *X*. Then, the result implies that the ratio λ_{1}(*G,X*)/μ_{1}(*G*) is bounded from below by a positive constant which is independent of the graph *G*. It follows that any isometric action of a random group of the graph model on such *X* has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.

#### Article information

**Source**

Kodai Math. J., Volume 39, Number 1 (2016), 129-153.

**Dates**

First available in Project Euclid: 22 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1458651696

**Digital Object Identifier**

doi:10.2996/kmj/1458651696

**Mathematical Reviews number (MathSciNet)**

MR3478275

**Zentralblatt MATH identifier**

1348.53051

#### Citation

Toyoda, Tetsu. Fixed point property for a CAT(0) space which admits a proper cocompact group action. Kodai Math. J. 39 (2016), no. 1, 129--153. doi:10.2996/kmj/1458651696. https://projecteuclid.org/euclid.kmj/1458651696