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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 handlebody group into the genus mapping class group is conjugation by a mapping class group element.
On the other hand, we construct an injection of the genus handlebody group into a genus mapping class group which is not conjugate into a handlebody group.
Cite this article
Sebastian Hensel, Rigidity and flexibility for handlebody groups. Comment. Math. Helv. 93 (2018), no. 2, pp. 335–358