Groups, Geometry, and Dynamics

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Volume 3, Issue 3, 2009, pp. 359–377
DOI: 10.4171/GGD/61

Published online: 2009-09-30

Copies of one-ended groups in mapping class groups

François Dahmani[1] and Koji Fujiwara[2]

(1) Université de Grenoble I, Saint-Martin-D'hères, France
(2) Kyoto University, Japan

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.

Keywords: Mapping class groups, pseudo-Anosov diffeomorphisms, curve complex

Dahmani François, Fujiwara Koji: Copies of one-ended groups in mapping class groups. Groups Geom. Dyn. 3 (2009), 359-377. doi: 10.4171/GGD/61