From Lang's conjecture to Torelli groups

Abstract: 

First we recall one of Lang's conjectures in diophantine geometry on the interplay between subvarieties and translated subgroups in a commutative algebraic group (proved by M. Laurent in the case of affine tori in 1984).

Then we present the technique of resonance and characteristic varieties, a powerful tool in the study of fundamental groups of algebraic varieties.

Finally, using the two ingredients above, we show that the Torelli groups $T_g$ have some surprising finiteness properties for. In particular, we show that for any subgroup containing the Johnson kernel, the complex vector space is finite dimensional.

All the details are available in our joint preprint with S. Papadima arXiv:1002.0673.