  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.
