We prove that the first complex homology of the Johnson subgroup of the Torelli group is a non-trivial, unipotent -module for all and give an explicit presentation of it as a \Sym_\dot H_1(T_g,\C)-module when . We do this by proving that, for a finitely generated group 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 . In this setup, we also obtain a precise nilpotence test.
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Stefan Papadima, Alexandru Dimca, Richard Hain, The abelianization of the Johnson kernel. J. Eur. Math. Soc. 16 (2014), no. 4, pp. 805–822DOI 10.4171/JEMS/447