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

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  20 October 2025

QUENTIN FAES ,
GWÉNAËL MASSUYEAU  and
MASATOSHI SATO
QUENTIN FAES
Affiliation:
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, CH-8057 Zurich, Switzerland. e-mail: quentin.faes@uclouvain.be
GWÉNAËL MASSUYEAU
Affiliation:
Université Bourgogne Europe, CNRS, IMB (UMR 5584), 21000 Dijon, France. e-mail: gwenael.massuyeau@ube.fr
MASATOSHI SATO
Affiliation:
Department of Mathematics and Data Science, Tokyo Denki University, 5 Senjuasahi-cho, Adachi-ku, Tokyo 120-8551, Japan. e-mail: msato@mail.dendai.ac.jp
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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.

MSC classification

Primary: 20F38: Other groups related to topology or analysis
Secondary: 20F12: Commutator calculus 20F14: Derived series, central series, and generalizations 20F34: Fundamental groups and their automorphisms

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , First View , pp. 1 - 39
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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