The algebraic structure of hyperbolic graph braid groups

Abstract

Genevois recently classified which graph braid groups on $\ge 3$ strands are word hyperbolic. In the $3$-strand case, he asked whether all such word hyperbolic groups are actually free; this reduced to checking two infinite classes of graphs: sun and pulsar graphs. We prove that $3$-strand braid groups of sun graphs are free. On the other hand, it was known to experts that $3$-strand braid groups of most pulsar graphs contain surface subgroups. We provide a simple proof of this and prove an additional structure theorem for these groups.

Figures (8)

• Figure 2.1: Let $X \subset \mathbb{R}^n$ be the above finite cube complex, and define a PL Morse function $f:X\to \mathbb R$ by projection to the $y$-axis. Then $\mathop{\mathrm{Lk}}\nolimits_{\downarrow}(v_0)$ is shown in blue, while $\mathop{\mathrm{Lk}}\nolimits_{\downarrow}(v_4)$ is empty. Let $J = [a, b]$. Then $f^{-1}(b)\cap X^{0}$ consists of the three vertices $v_1$, $v_2$, and $v_3$, and $f^{-1}(J)$ deformation retracts to the red space -- the union of the horizontal segment $f^{-1}(a)$ and the cones on the descending links of the three vertices.
• Figure 2.2: In the figure on the left, $Z \cup \text{Cone($v$,$\mathop{\mathrm{Lk}}\nolimits_{\downarrow}(v)$)}$ is the deformation retract of $f^{-1}([a,c])$ and is homotopy equivalent to $Z\cup {\bigvee_{\text{finite}}} S^1$ (on the right).
• Figure 2.3: The height function for the cycle graph $\Gamma$.
• Figure 2.4: The vertices $\{a_1,a_2,a_5\}, \{a_1,a_3,a_4\}$ and $\{a_2, a_4, a_6\}$ in $\mathrm{Conf}^\square_{3}({\Gamma})$ and their descending links.
• Figure 3.1: The sun graph $\mathcal{S}(3,1,3,2,0,2)$, where $n=6$. An admissible subdivision is obtained by subdividing each edge once (the even-labeled vertices are hence missing). For the inductive part of the proof of Theorem \ref{['t:sun']}, we will attach a ray to the top vertex of the cycle.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3
• proof
• Remark 2.4
• Example 2.5