Quasi-isometry classification of certain graph $2$-braid groups and its applications

TL;DR

This paper classifies certain graph 2-braid groups up to quasi-isometry using intersection complexes, and provides criteria and algorithms for identifying when these groups are quasi-isometric to right-angled Artin groups or each other.

Contribution

It introduces a classification method for graph 2-braid groups via intersection complexes and develops an algorithm for comparing 4-braid groups over trees.

Findings

Classified 2-braid groups over graphs with circumference ≤ 1 up to quasi-isometry.

Identified conditions for when such groups are quasi-isometric to right-angled Artin groups.

Provided an algorithm to determine quasi-isometry between 4-braid groups over trees.

Abstract

In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this paper, using the theory of intersection complexes, we classify the class of 2-braid groups over graphs with circumference $\leq 1$ up to quasi-isometry. Moreover, we find a sufficient condition when such a graph 2-braid group is quasi-isometric to a right-angled Artin group or not. Finally, by applying the same method, we also find that there is an algorithm to determine whether two 4-braid groups over trees are quasi-isometric or not.