Rigidity and Flexibility for Handlebody Groups

Authors :: Hensel, Sebastian
Publication Year :: 2016
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.<br />Comment: 21 pages, 4 figures