Almost quasi-isometries and more non-exact groups

TL;DR

This paper investigates the properties of almost quasi-isometries in the context of finitely generated random groups, demonstrating how these maps affect graph properties and expanding the class of known non-C*-exact groups.

Contribution

It extends Gromov's techniques to broader graph sequences, providing new examples of non-C*-exact groups through analysis of almost quasi-isometries.

Findings

Images of large girth graphs under almost quasi-isometries lack property A

Broader applicability of Gromov's methods to non-expander graph sequences

Identification of new finitely generated non-C*-exact groups

Abstract

We study permanence results for almost quasi-isometries, the maps arising from the Gromov construction of finitely generated random groups that contain expanders (and hence that are not C*-exact). We show that the image of a sequence of finite graphs of large girth and controlled vertex degrees under an almost quasi-isometry does not have Guoliang Yu's property A. We use this result to broaden the application of Gromov's techniques to sequences of graphs which are not necessarily expanders. Thus, we obtain more examples of finitely generated non-C*-exact groups.