Type: Seminar

Distribution: World

Expiry: 12 Mar 2010

Auth: jonh@asti.maths.usyd.edu.au

Alex Dimca is now at the University of Nice. (Many would remember him as a colleague at Sydney in the early 1990s!) From Lang’s Conjecture to finiteness properties of 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 $g>3$. In particular, we show that for any subgroup $N$ in $T_g$ containing the Johnson kernel $K_g$, the complex vector space $N_{ab} \otimes C$ is finite dimensional. All the details are available in our joint preprint with S. Papadima arXiv:1002.0673.