Class

Abstract

. For some g ≥ 3, let Γ be a finite index subgroup of the mapping class group of a genus g surface (possibly with boundary components and punc- tures). An old conjecture of Ivanov says that the abelianization of Γ should be finite. In the paper we prove two theorems supporting this conjecture. For the first, let T x denote the Dehn twist about a simple closed curve x . For some n ≥ 1, we have T nx ∈ Γ. We prove that T nx is torsion in the abelianization of Γ. Our second result shows that the abelianization of Γ is finite if Γ contains a “large chunk” (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves.