# The abelianization of the Johnson kernel

Alexandru Dimca; Richard Hain; Stefan Papadima

Journal of the European Mathematical Society (2014)

- Volume: 016, Issue: 4, page 805-822
- ISSN: 1435-9855

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topDimca, Alexandru, Hain, Richard, and Papadima, Stefan. "The abelianization of the Johnson kernel." Journal of the European Mathematical Society 016.4 (2014): 805-822. <http://eudml.org/doc/277466>.

@article{Dimca2014,

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 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.},

author = {Dimca, Alexandru, Hain, Richard, Papadima, Stefan},

journal = {Journal of the European Mathematical Society},

keywords = {Torelli group; Johnson kernel; Malcev completion; $I$-adic completion; characteristic variety; support; nilpotent module; arithmetic group; associated graded Lie algebra; infinitesimal Alexander invariant; Torelli group; Johnson kernel; Malcev completion; -adic completion; characteristic variety; support; nilpotent module; arithmetic group; associated graded Lie algebra; infinitesimal Alexander invariant},

language = {eng},

number = {4},

pages = {805-822},

publisher = {European Mathematical Society Publishing House},

title = {The abelianization of the Johnson kernel},

url = {http://eudml.org/doc/277466},

volume = {016},

year = {2014},

}

TY - JOUR

AU - Dimca, Alexandru

AU - Hain, Richard

AU - Papadima, Stefan

TI - The abelianization of the Johnson kernel

JO - Journal of the European Mathematical Society

PY - 2014

PB - European Mathematical Society Publishing House

VL - 016

IS - 4

SP - 805

EP - 822

AB - 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 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.

LA - eng

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

UR - http://eudml.org/doc/277466

ER -

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