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

TL;DR

This paper proves the finite generation of the Torelli group for surfaces with genus at least 3 and 2 boundary components, and also establishes finite generation of Johnson's kernel for genus at least 5.

Contribution

It establishes the finite generation of the Torelli group for specific surface types and answers a key question about stabilizer subgroups, advancing understanding of their algebraic properties.

Findings

Torelli group of genus ≥3 with 2 boundary components is finitely generated.

The stabilizer subgroup of a non-separating simple closed curve is finitely generated.

Johnson's kernel is finitely generated for genus ≥5.

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.