Problems on handlebody groups

TL;DR

This paper surveys recent developments and open problems in the study of handlebody groups, focusing on geometric, cohomological properties and their connections to automorphism groups of free groups.

Contribution

It provides an overview of constructions, open problems, and applies the cheap α-rebuilding property to analyze homology growth in handlebody groups.

Findings

Discussion of geometric and (co)homological properties

Application of cheap α-rebuilding property to homology growth

Identification of open problems and recent trends

Abstract

We survey a number of constructions and open problems related to the handlebody group, with a focus on recent trends in geometric group theory, (co)homological properties, and its relationship to outer automorphism groups of free groups. We also briefly describe how the \emph{cheap $\alpha$-rebuilding property} of Abert, Bergeron, Fraczyk, and Gaboriau can be applied using the disc complex to deduce results about the homology growth of the handlebody group.