Stable subgroups of the genus 2 handlebody group

Chesser, Marissa

doi:10.2140/agt.2022.22.919

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Stable subgroups of the genus $2$ handlebody group

Marissa Chesser

Algebraic & Geometric Topology 22 (2022) 919–971

DOI: 10.2140/agt.2022.22.919

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.

Keywords

hierarchically hyperbolic, handlebody group, stable subgroup, disk graph, CAT(0) cube complex

Mathematical Subject Classification

Primary: 20F65

Secondary: 20F67, 57M07

References

Publication

Received: 18 September 2020

Revised: 22 February 2021

Accepted: 28 March 2021

Published: 3 August 2022

Authors

Marissa Chesser
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States
https://sites.google.com/view/marissa-miller/home

Volume 22, issue 2 (2022)