Work

On the degree-two part of the associated graded of the lower central series of the Torelli group

Year: 2025

Type: article

Abstract: We consider the associated graded $\bigoplus_{k\geq 1} \Gamma_k \mathcal{I} /\Gamma_{k+1} \mathcal{I} $ of the lower central series $\mathcal{I}\,=\,\Gamma_1 \mathcal{I}\supset \Gamma_2 \mathcal{I}\supset \Gamma_3 \mathcal{I} \supset \cdots$ of the Torelli group $\mathcal{I}$ of a compact oriented surface. Its degree-one part is well understood by D. Johnson’s seminal works on the abelianization of the Torelli group. The knowledge of the degree-two part $(\Gamma_2 \mathcal{I} / \Gamma_3 \mathcal{I})\otimes \mathbb{Q}$ with rational coefficients arises from works of S. Morita on the Casson invariant and R. Hain on the Malcev completion of $\mathcal{I}$ . Here, we prove that the abelian group $\Gamma_2 \mathcal{I} / \Gamma_3 \mathcal{I}$ is torsion-free, and we describe it as a lattice in a rational vector space. As an application, the group $\mathcal{I}/\Gamma_3 \mathcal{I}$ is computed, and it is shown to embed in the group of homology cylinders modulo the surgery relation of $Y_3$ -equivalence.

Source: Mathematical Proceedings of the Cambridge Philosophical Society

DOI: 10.1017/s0305004125101539

Authors: QUENTIN FAES, GWÉNAËL MASSUYEAU, MASATOSHI SATO

Institutions: University of Zurich, Université de Bourgogne, Institut de Mathématiques de Bourgogne, Université Bourgogne Europe, Tokyo Denki University

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: Agence Nationale de la Recherche

Awards: Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité, Algebraic Aspects of Mapping Class Groups and Related Groups