Stable Subgroups of the Genus Two Handlebody Group

Abstract

We show that a finitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenstädt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Berlyne-Durham-Russell. We also present several applications of the main theorems, and show that the higher genus analogues of the genus two results do not hold.

arXiv comment45 pages, 11 figures. Changes made following referee suggestions. Updated author's last name (from Miller to Chesser)

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Cited by in corpus (4)

• Stable cubulations, bicombings, and barycenters
• Uniform difference between the length spectra of Out(F2) and the genus two handlebody group
• Extensions of multicurve stabilizers are hierarchically hyperbolic
• Problems on handlebody groups