Link Conditions for the Haagerup Property

TL;DR

This paper establishes a link condition in polygonal complexes that guarantees the automorphism group has the Haagerup property, with applications to groups acting on specific triangle complexes.

Contribution

It introduces a new link condition criterion for polygonal complexes ensuring the Haagerup property for their automorphism groups.

Findings

The link condition suffices for the Haagerup property in automorphism groups.

Application to automorphisms of triangle complexes with specific vertex links.

Demonstrates the property for groups acting properly on these complexes.

Abstract

We provide a condition on the links of a polygonal complex X that is sufficient to ensure Aut(X) has the Haagerup property, and hence so do any closed subgroups of Aut(X) (in particular, any group acting properly on X). We provide an application of this work by considering the group of automorphisms of simply-connected triangle complexes where the link of every vertex is isomorphic to the graph F090A, as constructed by \'Swi\k{a}tkowski.