One-ended subgroups of mapping class groups

Let Σ be a closed orientable surface, and write Map(Σ) for its mapping class group — the group of self-homeomorphisms up to homotopy. The Nielsen-Thurston classification partitions the non-trivial elements of Map(Σ) into finite order, reducible, and pseudoanosov, the last being the “generic” case. A subgroup of Map(Σ) is purely pseudoanosov if every non-trivial element is pseudoanosov. Such a subgroup is torsion-free. We shall say that a group is indecomposable if and only if it does not split as a free product. It is a theorem of Stallings that a torsion-free finitely generated group is indecomposable if and only if it is one-ended. It is an open question as to whether Map(Σ) can contain any one-ended finitely generated purely pseudoanosov subgroup. Indeed the only purely pseudoanosov subgroups known at present are all free. (See the surveys [Re] and [Mo].) In this paper, we show that there can be only finitely many conjugacy classes of finitely presented one-ended subgroups of a given isomorphism type. More precisely: