The abelianization of the Johnson kernel

作者：Dimca Alexandru^*; Hain Richard; Papadima Stefan

来源：Journal of the European Mathematical Society, 2014, 16(4): 805-822.

DOI：10.4171/JEMS/447

• Torelli group
• Johnson kernel
• Malcev completion
• I -adic completion
• characteristic variety
• support
• nilpotent module
• arithmetic group
• associated graded Lie algebra
• infinitesimal Alexander invariant

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 >= 4 and give an explicit presentation of it as a Sym(center dot) H-1 (T-g, C)-module when g >= 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 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.

• 出版日期2014