The abelianization of the Johnson kernel

TL;DR

This paper investigates the first homology of the Johnson kernel within the Torelli group, revealing its structure as a unipotent module and providing explicit presentations for higher genus cases.

Contribution

It establishes the non-triviality of the first homology as a unipotent module and offers an explicit presentation for genus g ≥ 6, linking homology to the infinitesimal Alexander invariant.

Findings

First homology is a non-trivial unipotent module for all g ≥ 4.

Explicit module presentation provided for g ≥ 6.

Introduces a nilpotence test based on characteristic varieties.

Abstract

We prove that the first complex homology of the Johnson subgroup of the Torelli group $T_g$ is a non-trivial unipotent $T_g$-module for all $g\ge 4$ and give an explicit presentation of it as a $\Sym H_1(T_g,\C)$-module when $g\ge 6$. We do this by proving that, for a finitely generated group $G$ satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel $K$ is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of $G$. In this setup, we also obtain a precise nilpotence test.