CAT(0) spaces on which a certain type of singularity is bounded

TL;DR

This paper establishes a geometric condition for certain CAT(0) spaces with bounded singularities, ensuring their invariants are uniformly bounded, which implies fixed-point properties for related random groups.

Contribution

The paper introduces a new geometric condition that guarantees uniform bounds on Izeki-Nayatani invariants for a class of CAT(0) spaces with bounded singularities.

Findings

Izeki-Nayatani invariants are uniformly bounded below 1 under the new condition.

Spaces with this condition include Gromov's bounded singularities spaces.

Random groups in Gromov's model have fixed-point properties for these spaces.

Abstract

In this paper, we present a geometric condition for a family of CAT(0) spaces, which ensures that the Izeki-Nayatani invariants of spaces in the family are uniformly bounded from above by a constant strictly less than 1. Each element of such a family with this condition is a space presented by M. Gromov as an example of a "CAT(0) space with "bounded" singularities". Combining our result with a result of Izeki, Kondo and Nayatani, we see that random groups of Gromov's graph model have a fixed-point property for such a family.