Problems on handlebody groups
Naomi Andrew, Sebastian Hensel, Sam Hughes, Richard D. Wade
TL;DR
The paper surveys open problems on the handlebody group $\mathrm{Mod}(V_g)$, emphasizing its connections to $\mathrm{Out}(F_g)$ and $\mathrm{Mod}(S_g)$. It develops and applies the disc complex $\mathcal{D}(V)$ and handlebody horoballs to model actions and study homology growth through the cheap $\alpha$-rebuilding property, yielding vanishing results for $\ell^2$-Betti numbers. A central theme is the interaction between large-scale geometry, (co)homology, and profinite aspects, including congruence properties, goodness, and profinite rigidity. The work outlines concrete conjectures and open questions—ranging from universal acylindrical actions and QI-rigidity to lifting problems from $\mathrm{Out}(F_g)$$—providing a broad program for future research with clear geometric and algebraic directions.
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 $α$-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.