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

TL;DR

The article proves that the Torelli group for a genus $g\ge 3$ surface with two boundary components is finitely generated, answering a question of Putman about stabilizers of nonseparating curves. It develops a cubic-size generating set by decomposing the surface into manageable subsurfaces and employing chains of bounding pair maps; it also proves finite generation of the Johnson kernel $\mathcal{K}(\Sigma,S)$ for $g\ge 5$ using a BNS-invariant approach. Core techniques include a detailed analysis of chain maps, Birman exact sequences, and a structured $\boldsymbol{n}$-group framework with associated commuting graphs and Zariski-irreducibility arguments. The work extends prior results for one-boundary cases to two boundaries, linking classical Torelli theory with modern finiteness machinery and yielding explicit generators and new structural insights. The findings have significance for understanding the algebraic structure of mapping class groups and their Torelli subgroups in more intricate boundary configurations.

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.

Figures (8)

• Figure 1: Model of subsurfaces that is used here
• Figure 2: The action of a Dehn twist $T_c$.
• Figure 3: The mapping class group is generated by Dehn twists about these curves.
• Figure 4: Standard generators for $H_g$, and $H_1^{P'}(\Sigma_{g, 2})$.
• Figure 5: Lantern formed by P'-bounding pairs and P'-separating curve.

Theorems & Definitions (46)

• Theorem 1.1: Putman
• Theorem A
• Corollary 1.2
• Theorem B
• Theorem C
• Theorem 2.1: Putman
• Lemma 2.2
• proof
• Lemma 2.3
• proof