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 cheap α-rebuilding property of Abert, Bergeron, Fraczyk and Gaboriau (J. Eur. Math. Soc. 27, 2293–2357. (doi:10.4171/jems/1411)) can be applied using the disc complex to deduce results about the homology growth of the handlebody group.