A note on the abelianizations of finite-index subgroups of the mapping class group

Let Σ g,b be an oriented genus g surface with b boundary components and p punctures and let Mod(Σ g,b) be its mapping class group, that is, the group of isotopy classes of orientation–preserving diffeomorphisms of Σ g,b that fix the boundary components and punctures pointwise (we will omit b or p if they vanish). A long–standing conjecture of Ivanov (see [7] for a recent discussion) says that for g ≥ 3, the group Mod(Σ g,b) does not virtually surject onto Z. In other words, if Γ is a finite-index subgroup of Mod(Σ g,b), then H1(Γ;Q) = 0. The goal of this note is to offer some evidence for this conjecture. If G is a group and g∈G, then we will denote by [g]G the corresponding element of H1(G;Q). Also, for a simple closed curve γ on Σ g,b, we will denote by Tγ the corresponding right Dehn twist. Observe that if Γ is any finite-index subgroup of Mod g,b, then T n γ ∈ Mod p g,b for some n ≥ 1. Our first result is the following. Theorem 1.1 (Powers of twists vanish). For some g ≥ 3, let Γ < Mod(Σ g,b) satisfy [Mod(Σ p g,b) : Γ] < ∞ and let γ be a simple closed curve on Σ g,b. Pick n ≥ 1 such that T n γ ∈ Γ. Then [T n γ ]Γ = 0.