Stable Subgroups of the Genus Two Handlebody Group
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Publication:6348853
DOI10.2140/AGT.2022.22.919arXiv2009.05067MaRDI QIDQ6348853
Publication date: 10 September 2020
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.
Geometric group theory (20F65) Topological methods in group theory (57M07) Hyperbolic groups and nonpositively curved groups (20F67)
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