Stable subgroups of the genus 2 handlebody group

Abstract

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.