Finiteness properties of the Torelli group of surfaces with 2 boundary components

Abstract

In this paper we prove that the Torelli group of a surface of genus at least 3 with 2 boundary components is finitely generated. As a consequence, we answer Putman's question on the finite generation of the stabilizer subgroup of the Torelli group of a non separating simple closed curve. Furthermore, we prove that the Johnson's kernel is finitely generated if the genus of the surface is at least 5.