Arithmetic group symmetry and finiteness properties of Torelli groups

Details Arithmetic group symmetry and finiteness properties of Torelli groups Arithmetic group symmetry and finiteness properties of Torelli groups

2010-02-03

arxiv.org/abs/1002.0673v3

Mathematics

Group Theory

Compared to the previous version, Theorem A (2) and Theorem D (5) are proved in stronger forms

Scientific paper

We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have finite first Betti number, when the genus is at least four. We also prove that, in this range, the $I$-adic completion of the Alexander invariant is finite-dimensional, and the Kahler property for the Torelli group implies the finite generation of the Johnson kernel.

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