On certain subspaces of $2$-configuration spaces of graphs

TL;DR

The work advances understanding of graph braid groups by exploiting the cubical geometry of discrete configuration spaces UD_n(Γ). It achieves a complete freeness classification and develops the UP_2(Γ) framework together with intersection complexes to capture quasi-isometry types, establishing both RAAG-like and non-RAAG behavior among infinitely many 2-braid groups. By introducing bunches of grapes and the Y^max-hierarchy, the authors connect geometric decomposition to algebraic invariants, enabling precise RAAG QI results and new examples of relative hyperbolicity. The results illuminate when graph braid groups can be modeled by RAAGs, reveal new large-scale geometric phenomena, and provide tools for ongoing classification of graph braid groups up to quasi-isometry with RAAGs. Overall, the paper deepens the link between cubical geometry, intersection patterns, and the large-scale geometry of graph braid groups with broad implications for geometric group theory.

Abstract

We investigate the large-scale geometry of graph braid groups \(\bbB_n(\graf)\) using the cubical structure of their discrete configuration spaces \(UD_n(\graf)\). First, we give a complete classification of when a graph \(n\)-braid group is free, using only cubical methods and without appealing to discrete Morse theory. We then focus on graph \(2\)-braid groups and study the square complex \(UD_2(\graf)\), which is special in the sense of Haglund--Wise, via its maximal product subcomplexes and intersection complex introduced in \cite{Oh22}. Under natural connectivity and embeddedness assumptions, we show that the union \(UP_2(\graf)\) of maximal product subcomplexes of \(UD_2(\graf)\) captures essential quasi-isometry information of \(\bbB_2(\graf)\), the fundamental group of \(UD_2(\graf)\). Applying this framework to an infinite family of graphs, we obtain infinitely many graph \(2\)-braid groups that are quasi-isometric to right-angled Artin groups and infinitely many that are not, extending the examples of \cite{Oh22}, and we indicate new phenomena in relative hyperbolicity.

Figures (14)

• Figure 1: Examples of $Y$'s having disconnected $\mathcal{I}(\overline{{Y}^\mathsf{max}})$
• Figure 2: The union of five tori with laces
• Figure 3: (Counter)examples in $\mathcal{Y}_{(i)}$. All edges are contained in $2$-cubes except in (A), where the $4$-cycle on the right does not bound a $2$-cubes.
• Figure 4: Examples of $UD_2(\mathsf{\Gamma})$.
• Figure 5: Example and non-example of graphs with connected curve graphs

Theorems & Definitions (165)

• Theorem 1.1: \ref{['theorem:freeness classification']}
• Theorem 1.2
• Proposition 1.3: \ref{['lem:Connected']}
• Proposition 1.4: \ref{['Lem:SPSinGBG', 'corollary:maximal']}
• Theorem 1.5: \ref{['Thm:SubclassdefiningQII']}
• Definition 1.6: Bunches of grapes, \ref{['Def:BunchesofGrapes']}
• Proposition 1.7: \ref{['Prop:free factor']}
• Theorem 1.8: \ref{['theorem:QIbetweenGBGs']}
• Theorem 1.9: \ref{['theorem:isomorphictoRAAG']}
• Theorem 1.10: \ref{['theorem:NotQItoRAAG', 'theorem:necessary condition 2']}