Stable Subgroups of the Genus Two Handlebody Group

Marissa Chesser

doi:10.2140/agt.2022.22.919

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arXiv · 2009.05067

Stable Subgroups of the Genus Two Handlebody Group

Marissa Chesser

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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 Hamenst\"adt-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.

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Marissa Chesser. 2020-09-10. Stable Subgroups of the Genus Two Handlebody Group. https://doi.org/10.2140/agt.2022.22.919

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