Copies of one-ended groups in mapping class groups

Abstract

We establish that, given Σ a compact orientable surface and G a finitely presented one-ended group, the set of copies of G in the mapping class group MCG(Σ) consisting of only pseudo-Anosov elements except identity is finite up to conjugacy. This relies on a result of Bowditch on the same problem for images of surfaces groups. He asked us whether we could reduce the case of one-ended groups to his result; this is a positive answer. Our work involves analogues of Rips and Sela’s canonical cylinders in curve complexes and an argument of Delzant to bound the number of images of a group in a hyperbolic group.