Suppose we have a finitely presented one-ended group with a purely loxodromic action on a Gromov hyperbolic space satisfying an acylindricity condition. We show that given a generating set, there is an automorphism of the group, and some point in the space which is moved a bounded distance by each of the images of the generators under the automorphism. Here the bound depends only on the group, generating set, and constants of hyperbolicity and acylindricity. With results from elsewhere, this implies that, up to conjugacy, there can only be finitely many purely pseudoanosov subgroups of a mapping class group that are isomorphic to a given finitely presented one-ended group.
In ``Hyperbolic geometry and geometric group theory'' (eds. K.Fujiwara, S.Kojima, K.Ohshika) Advanced Studies in Pure Mathematics, Mathematical Society of Japan (2017) 13-36.