Quasi-isometry invariants of weakly special square complexes

TL;DR

This paper introduces the intersection complex as a new quasi-isometry invariant for weakly special square complexes and applies it to classify certain 2-dimensional right-angled Artin groups and graph braid groups.

Contribution

It defines the intersection complex and demonstrates its effectiveness in the quasi-isometric classification of specific groups.

Findings

Intersection complex is a quasi-isometry invariant.

Classifies 2D right-angled Artin groups with specific properties.

Identifies infinitely many graph braid groups quasi-isometric to RAAGs.

Abstract

We define the intersection complex for the universal cover of a compact weakly special square complex and show that it is a quasi-isometry invariant. By using this quasi-isometry invariant, we study the quasi-isometric classification of 2-dimensional right-angled Artin groups and planar graph 2-braid groups. Our results cover two well-known cases of 2-dimensional right-angled Artin groups: (1) those whose defining graphs are trees and (2) those whose outer automorphism groups are finite. Finally, we show that there are infinitely many graph 2-braid groups which are quasi-isometric to right-angled Artin groups and infinitely many which are not.