Problems on handlebody groups

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.

20 pages, comments welcome!

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