Rigidity and Flexibility for Handlebody Groups

TL;DR

This paper proves that finite index subgroups of handlebody groups are rigid within their ambient mapping class groups, but also constructs non-conjugate embeddings into larger genus mapping class groups.

Contribution

It establishes rigidity results for handlebody groups and demonstrates the existence of non-conjugate embeddings into higher genus mapping class groups.

Findings

Finite index subgroups are rigid in their ambient groups.

Any injective map of a handlebody subgroup is conjugate to a mapping class group element.

Existence of non-conjugate embeddings into larger genus mapping class groups.

Abstract

We show that finite index subgroups of the handlebody group are rigid in their ambient mapping class group: any injective map of a finite index subgroup of the genus $g$ handlebody group into the genus $g$ mapping class group is conjugation by a mapping class group element. On the other hand, we construct an injection of the genus $g$ handlebody group into a genus $h>g$ mapping class group which is not conjugate into a handlebody group.