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

Abstract

We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, 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 $\mathbb{B}_n(\mathsf{\Gamma})$ 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 $UD_2(\mathsf{\Gamma})$ and the intersection complex introduced in \cite{Oh22}, we show that, under natural assumptions, their union captures essential quasi-isometry information about $\mathbb{B}_2(\mathsf{\Gamma})$. 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.