Manifold models for hyperbolic graph braid groups on three strands

Abstract

Given a finite graph $Γ$, the associated graph braid group $B_n(Γ)$ is the fundamental group of the unordered $n$-point configuration space of $Γ$. 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 $B_3(Θ_m)$, where $Θ_m$ is the generalized $Θ$-graph, a suspension of $m$-points. We show that $B_3(Θ_5)$ is a $3$-manifold group while $B_3(Θ_m)$ is not even quasi-isometric to a $3$-manifold group for $m \geq 7$.

Figures (10)

• Figure 1: The tripod $Y$ (a cone on 3 points) and $\mathop{\mathrm{Conf}}\nolimits_{2}(Y)$.
• Figure 2: The families of graphs $\Gamma$ for which $B_3(\Gamma)$ is hyperbolic.
• Figure 3: The $\Theta_5$ graph.
• Figure 4: Links of $ijk$, $aij$, and $abi$ in $\mathrm{Conf}_{3}(\Theta_{5})$.
• Figure 5: The underlying graph $X'$ for $\mathrm{Conf}_{3}(\Theta_{5})$.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: MR1621973
• Definition 2.2: MR1621973
• Theorem 2.3
• Remark
• proof : Proof of Theorem \ref{['thm:main5']}
• Remark
• Definition 3.1
• Theorem 3.2