We show that a finitely generated subgroup of the genus 2 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 2 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 2 handlebody by appealing to a construction of
Hamenstädt and Hensel. We then utilize the characterization of stable
subgroups of hierarchically hyperbolic groups provided by Abbott, Behrstock,
Berlyne, Durham and Russell. We also present several applications of the main
theorems, and show that the higher-genus analogues of the genus 2 results do not
hold.
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