JournalsjemsVol. 16, No. 4pp. 805–822

The abelianization of the Johnson kernel

  • Stefan Papadima

    Romanian Academy, Bucharest, Romania
  • Alexandru Dimca

    Université de Nice Sophia Antipolis, France
  • Richard Hain

    Duke University, Durham, USA
The abelianization of the Johnson kernel cover
Download PDF

Abstract

We prove that the first complex homology of the Johnson subgroup of the Torelli group TgT_g is a non-trivial, unipotent TgT_g-module for all g4g\ge 4 and give an explicit presentation of it as a \Sym_\dot H_1(T_g,\C)-module when g6g\ge 6. We do this by proving that, for a finitely generated group GG satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel is a nilpotent module over the corresponding Laurent polynomial ring, isomorphic to the infinitesimal Alexander invariant of the associated graded Lie algebra of GG. In this setup, we also obtain a precise nilpotence test.

Cite this article

Stefan Papadima, Alexandru Dimca, Richard Hain, The abelianization of the Johnson kernel. J. Eur. Math. Soc. 16 (2014), no. 4, pp. 805–822

DOI 10.4171/JEMS/447