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

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 $\mathcal{MC}\mathcal{G}(\Sigma )$ 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.

16 pages, 3 figures, revised

Cited by in corpus (2)

• Median structures on asymptotic cones and homomorphisms into mapping class groups
• The geometry of the curve graph of a right-angled Artin group