Surface subgroups of Mapping Class groups

Mod(Σ) = Homeo(Σ)/Homeo0(Σ), where, Homeo(Σ) is the group of orientation preserving homeomorphisms of Σ and Homeo0(Σ) are those homeomorphisms isotopic to the identity. When Σ is a closed surface of genus g ≥ 1 then we denote Mod(Σ) by Γg. In this case it is well-known that Γg is isomorphic to a subgroup of index 2 in Out(π1(Σ)). When g = 1, the subgroup structure of Γ1 is well-understood since it is simply the group SL(2,Z). When g ≥ 2, attempts to understand the subgroup structure of Γg have been made by exploiting the many analogies between Γg and non-uniform lattices in Lie groups. In particular, both the finite and infinite index subgroup structure of Γg has many parallels in the theory of lattices. For example, the question of whether Property T holds for Γg ≥ 3 (it fails for g = 2 by [29]), whether Γg has a version of the Congruence Subgroup Property, or towards the other extreme, whether there are finite index subgroups of Γg that surject onto Z (see for example [11], [16], [18] and [29] for more on these directions). The discussion and questions raised in this paper are motivated by analogies between the subgroup structure of Γg and non-cocompact but finite co-volume Kleinian groups. To that end, we are particularly interested in the nature of surface subgroups of Γg. For an exploration of other analogies between Γg and Kleinian groups see [13], [19] and [33]. Throughout, the terms surface (sub)group will be reserved for π1(Σg) where g ≥ 2, and a subgroup of Γg is said to be purely pseudo-Anosov if all non-trivial elements are pseudo-Anosov. Simply put our motivation is the following question.