The abelianization of the Johnson kernel

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.

19 pages, second version, to appear in JEMS

References in corpus (1)

• The Johnson homomorphism and its kernel

Cited by in corpus (4)

• Vanishing resonance and representations of Lie algebras
• The Johnson homomorphism and its kernel
• On the structure of the top homology group of the Johnson kernel
• Generating the Johnson filtration II: finite generation