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

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. Let Σ be a compact orientable surface (possibly with boundary components). The Mapping Class Group of Σ, denoted by MCG(Σ) is the group of isotopy classes of orientation preserving self-homeomorphisms of Σ. The aim of this paper is to report on a control on the family of subgroups of Mapping Class Groups that are isomorphic to a given finitely presented one-ended group. Definition 0.1 (Purely pseudo-Anosov) A subgroup of MCG(Σ) is said purely pseudoAnosov if all its elements, except the identity, are pseudo-Anosov mapping classes. A morphism of a group in MCG(Σ) is said purely pseudo-Anosov if it is injective and has purely pseudo-Anosov image. Up to now, the only known purely pseudo-Anosov subgroups of Mapping Class Groups are free. Recently, Brian Bowditch [4] has established the finiteness of the set of such images of surface groups, up to conjugacy. He uses deep results, in particular in 3-manifold geometry, and doing so, he proves a general powerful result [4, Proposition 8.1]. He asked us, however, whether it is possible to adapt the situation of an arbitrary finitely presented one-ended group to the setting of his Proposition. We provide here an affirmative answer. Theorem 0.2 Given Σ a compact orientable surface, and G a finitely presented one-ended group, there is a number N such that any purely pseudo-Anosov subgroup of MCG(Σ) isomorphic to G admits a set of generators (γi) for which there are a vertex v in the curve complex CC(Σ) with d(γiv, v) ≤ N , and a presentation on this set of generators with at most N relators, each of them of length at most N as words. This allows to apply the following important Bowditch’s result: The first author acknowledges partial support from the ANR grant ANR-06-JCJC-0099 The second author is partially supported by Grant-in-Aid for Scientific Research (No. 19340013).