On the Torelli Lie algebra

TL;DR

This paper investigates the Malcev Lie algebra of the Torelli group, establishing its Koszul property and characterizing the kernel of the Johnson homomorphism in terms of trivial symplectic representations.

Contribution

It proves that the Malcev Lie algebra of the Torelli group is Koszul and describes the kernel of the Johnson homomorphism in terms of trivial symplectic representations.

Findings

Malcev Lie algebra of Torelli group is Koszul

Kernel of Johnson homomorphism consists only of trivial symplectic representations

Results hold stably for surfaces of genus g

Abstract

We prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus $g$: stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial $Sp_{2g}(\mathbb{Z})$-representations lying in the centre.