Volume 22, issue 2 (2022)

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Stable subgroups of the genus $2$ handlebody group

Marissa Chesser

Algebraic & Geometric Topology 22 (2022) 919–971

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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hierarchically hyperbolic, handlebody group, stable subgroup, disk graph, CAT(0) cube complex
Mathematical Subject Classification
Primary: 20F65
Secondary: 20F67, 57M07
Received: 18 September 2020
Revised: 22 February 2021
Accepted: 28 March 2021
Published: 3 August 2022
Marissa Chesser
Department of Mathematics
University of Illinois at Urbana-Champaign
Urbana, IL
United States