  We establish that, given $\Sigma$ a compact orientable surface, and $G$ a
finitely presented one-ended group, the set of copies of $G$ in the mapping
class group $\mathcal{MCG}(\Sigma)$ 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.
