Two 2-braid groups over circumference-one graphs are quasi-isometric exactly when their quasi-minimal representatives are isometric, and the comparison is algorithmic.
Hierarchical hyperbolicity of graph products and graph braid groups
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Quasi-isometry classification of certain graph $2$-braid groups and its applications
Two 2-braid groups over circumference-one graphs are quasi-isometric exactly when their quasi-minimal representatives are isometric, and the comparison is algorithmic.