Superrigid subgroups and syndetic hulls in solvable Lie groups

Abstract

This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete subgroups D of connected Lie groups G, such that the homomorphisms defined on D can ("virtually") be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that D has this property if it is Zariski dense. The key ingredient is a result on the existence of syndetic hulls.

17 pages. This is the final version that will appear in the volume "Rigidity in Dynamics and Geometry," edited by M. Burger and A. Iozzi (Springer, 2002)