Properties of 3-strand graph braid groups for sun and pulsar graphs

Determine, for finite sun graphs and pulsar graphs Γ, the group-theoretic status of the 3-strand graph braid group B_3(Γ) = π1(Conf_3(Γ)): specifically, decide whether B_3(Γ) is free; decide whether B_3(Γ) is one-ended; decide whether B_3(Γ) is a surface group; and decide whether B_3(Γ) is a 3-manifold group.

Background

Graph braid groups B_n(Γ) arise as the fundamental groups of configuration spaces of n unordered points on a finite graph Γ. Genevois classified when such groups are word hyperbolic and identified that, in the 3-strand case, hyperbolicity occurs only for trees, rose graphs, sun graphs, and pulsar graphs. It was known that for trees and rose graphs certain properties (e.g., freeness) hold, prompting the broader inquiry for the remaining hyperbolic cases.

The present paper answers parts of this inquiry: it proves B_3(Γ) is free for sun graphs, shows many pulsar cases are not one-ended (via a free product decomposition when rays are present), and computes Euler characteristics for generalized theta graphs Θ_m, implying for m>7 that B_3(Θ_m) is not a 3-manifold group. It also shows Conf_3^□(Θ_4) is a closed, orientable surface of genus 3. The residual question concerns the full classification across sun and pulsar graphs regarding freeness, one-endedness, surface group status, and 3-manifold group status.

This question, originally posed by Genevois (Question 5.3 in his work), is restated explicitly by the authors to motivate their contributions and identify remaining gaps.