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

TL;DR

This paper proves that for a given one-ended finitely presented group, there are only finitely many conjugacy classes of its embeddings into a mapping class group that consist solely of pseudo-Anosov elements, extending Bowditch's results.

Contribution

It reduces the problem of classifying such embeddings to Bowditch's surface group case, using analogues of canonical cylinders and Delzant's argument.

Findings

Finiteness of conjugacy classes of certain group embeddings into mapping class groups.

Extension of Bowditch's results to one-ended groups.

Application of Rips and Sela techniques in the context of curve complexes.

Abstract

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.