The asymptotic geometry of right-angled Artin groups, I

TL;DR

This paper investigates the large-scale geometric properties of atomic right-angled Artin groups, establishing conditions under which they are quasi-isometric only if they are structurally identical, thus revealing a form of rigidity.

Contribution

It introduces the class of atomic right-angled Artin groups and proves that they are quasi-isometrically rigid, with quasi-isometry implying isomorphism.

Findings

Atomic right-angled Artin groups are not fully quasi-isometrically rigid.

A weaker form of rigidity holds for these groups.

Two atomic groups are quasi-isometric if and only if they are isomorphic.

Abstract

We study atomic right-angled Artin groups -- those whose defining graph has no cycles of length less than five, and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic.