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arXiv · 2602.15636

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

Byung Hee An·Sangrok Oh
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Abstract

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

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Byung Hee An, Sangrok Oh. 2026-03-24. On the large-scale geometry of graph braid groups via cubical structures. https://arxiv.org/abs/2602.15636

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