Kodai Mathematical Journal

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

Tetsu Toyoda

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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, μ1(G) be the linear spectral gap of G, and λ1(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 λ1(G,X)/μ1(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


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