Manifold models for hyperbolic graph braid groups on three strands

TL;DR

This paper investigates the geometric and topological properties of certain graph braid groups, specifically those arising from generalized Theta graphs, and determines which of these groups are 3-manifold groups.

Contribution

It provides a partial classification of when the braid groups on three strands over generalized Theta graphs are 3-manifold groups, identifying specific cases with this property.

Findings

B_3(Θ_5) is a 3-manifold group

B_3(Θ_m) for m ≥ 7 is not quasi-isometric to any 3-manifold group

The paper offers insights into the hyperbolic and manifold properties of these groups

Abstract

Given a finite graph $\Gamma$, the associated graph braid group $B_n(\Gamma)$ is the fundamental group of the unordered $n$-point configuration space of $\Gamma$. Genevois classified which graph braid groups are Gromov hyperbolic and asked the question: When do these groups arise as $3$-manifold groups? In this paper, we give a partial answer for $B_3(\Theta_m)$, where $\Theta_m$ is the generalized $\Theta$-graph, a suspension of $m$-points. We show that $B_3(\Theta_5)$ is a $3$-manifold group while $B_3(\Theta_m)$ is not even quasi-isometric to a $3$-manifold group for $m \geq 7$.