Stable Subgroups of the Genus Two Handlebody Group

TL;DR

This paper characterizes stable subgroups of the genus two handlebody group using hierarchical hyperbolicity and quasi-isometric embeddings into the disk graph, revealing new geometric insights and limitations for higher genus cases.

Contribution

It proves the genus two handlebody group is hierarchically hyperbolic and characterizes stable subgroups via quasi-isometric orbit maps, extending understanding of geometric group theory.

Findings

Stable subgroups correspond to quasi-isometric embeddings into the disk graph.

Genus two handlebody group is hierarchically hyperbolic.

Higher genus analogues of these results do not hold.

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.