On the large-scale geometry of graph braid groups via cubical structures

Abstract

We study the large-scale geometry of graph braid groups ${\mathbb{B}}_n(\hat{\mathsf{I}}“)$, viewed as the fundamental groups of discrete configuration spaces $U{D}_n(\hat{\mathsf{I}}“)$, which are special cube complexes in the sense of Haglund--Wise. Exploiting this cubical structure, we relate hyperbolicity, undistorted surface subgroups, and group-theoretic decompositions. As a consequence, we obtain a complete classification of when ${\mathbb{B}}_n(\hat{\mathsf{I}}“)$ is quasi-isometric to a free group via a purely geometric argument independent of discrete Morse theory. We then focus on graph $2$-braid groups. Using maximal product subcomplexes of $U{D}_2(\hat{\mathsf{I}}“)$ and the intersection complex introduced in \cite{Oh22}, we show that, under natural assumptions, their union captures essential quasi-isometry information about ${\mathbb{B}}_2(\hat{\mathsf{I}}“)$. As applications, we construct infinitely many graph $2$-braid groups that are quasi-isometric to right-angled Artin groups and infinitely many that are not, extending \cite{Oh22}, and we exhibit new phenomena in relative hyperbolicity.

v3: Revised title and improved exposition; minor clarifications throughout