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

Byung Hee An, Sangrok Oh

TL;DR

This work advances the quasi-isometric classification of graph $2$-braid groups by introducing the intersection complex as a coarse invariant and the framework of bunches of grapes. It proves that, for graphs with circumference $ ext{≤}1$, the $2$-braid groups admit an algorithmic quasi-isometry test via reductions to quasi-minimal representatives, with the intersection complex (and its reduced version) capturing the essential coarse geometry. The paper also develops a suite of graph-operations that preserve quasi-isometry, yielding rigidity results: two $2$-braid groups are quasi-isometric iff their quasi-minimal grape representatives are isometric. Applications include a refined RAAG-dichotomy for graph $2$-braids, a complete quasi-isometric classification of tree $4$-braid groups, and algorithms to decide these quasi-isometry relations, thereby linking configuration-space geometry to RAAGs and tree-gluing phenomena.

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.

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

TL;DR

This work advances the quasi-isometric classification of graph -braid groups by introducing the intersection complex as a coarse invariant and the framework of bunches of grapes. It proves that, for graphs with circumference , the -braid groups admit an algorithmic quasi-isometry test via reductions to quasi-minimal representatives, with the intersection complex (and its reduced version) capturing the essential coarse geometry. The paper also develops a suite of graph-operations that preserve quasi-isometry, yielding rigidity results: two -braid groups are quasi-isometric iff their quasi-minimal grape representatives are isometric. Applications include a refined RAAG-dichotomy for graph -braids, a complete quasi-isometric classification of tree -braid groups, and algorithms to decide these quasi-isometry relations, thereby linking configuration-space geometry to RAAGs and tree-gluing phenomena.

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 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.

Paper Structure

This paper contains 34 sections, 79 theorems, 95 equations, 23 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Bunches of grapes and their 2-braid groups
  3. Intersection complexes
  4. Operations on bunches of grapes
  5. The completeness of the invariant
  6. Applications
  7. Graph $2$-braid groups and RAAGs
  8. Tree $4$-braid groups
  9. Remarks and questions
  10. Organization of the paper
  11. Preliminaries
  12. Weakly special cube complexes
  13. Standard product subcomplexes
  14. (Reduced) Intersection complexes
  15. Free products of graphs
  16. ...and 19 more sections

Key Result

Lemma 2.7

Let $Y$ be an $n$-dimensional cube complex and let $\Gamma_1,\dots,\Gamma_n$ be 1-dimensional graphs. Then any immersion $\iota:\Gamma_1\times\dots\times\Gamma_n\to Y$ is a local isometry.

Figures (23)

  • Figure 1: Examples of a bunch of graphs and its stem
  • Figure 2: A part of $\overline{*_{i=1}^5\{(\Gamma_i,v_i)\}}$; at each vertex, the small bar indicates the universal cover $\overline{\Gamma}_i$.
  • Figure 3: Free product of graphs as an $m$-ary operation and as a composition of binary operations.
  • Figure 4: Examples of $UD_2(\Gamma)$.
  • Figure 5: An example of graph-of-groups decompositions $\mathcal{G}(\mathbb{B}_2(\Gamma,v))$ and $\mathcal{G}(\pi_1(UP_2(\Gamma,v)))$.
  • ...and 18 more figures

Theorems & Definitions (172)

  • Remark 2.1
  • Example 2.2: Graphs
  • Remark 2.3
  • Example 2.4: Salvetti complex
  • Example 2.5: Discrete $2$-configuration spaces of graphs
  • Lemma 2.7
  • proof
  • Example 2.8: Flats in $\mathop{\mathrm{CAT}}\nolimits(0)$ cube complexes
  • Definition 2.9: HW08Hua(b) (Weakly) special cube complex
  • Theorem 2.10: Hua(b), Theorem 1.3
  • ...and 162 more