Copies of a one-ended group in a Mapping Class Group

We establish that, given $\mathrm{\Sigma }$ a compact orientable surface, and $G$ a finitely presented one-ended group, the set of copies of $G$ in the mapping class group $\mathcal{M}\mathcal{C}\mathcal{G}(\mathrm{\Sigma })$ 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 canonical cylinders in curve complexes, and the argument of Delzant to bound the number of images of a one-ended group in a hyperbolic group.