Small cancellation labellings of some infinite graphs and applications

TL;DR

This paper develops small cancellation labellings for infinite graphs to construct groups with unique properties, including coarsely non-amenable groups that embed into Hilbert spaces and groups with embedded expanders.

Contribution

It introduces a novel method for constructing groups with exotic properties using small cancellation labellings on infinite graphs.

Findings

First examples of coarsely non-amenable groups that embed into Hilbert space

Groups with isometrically embedded expanders in their Cayley graphs

Proper actions on CAT(0) cubical complexes

Abstract

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: - We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu's Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. - We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs - in contrast,in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.