SUPERRIGIDITY OF LATTICES IN SOLVABLE LIE-GROUPS

Let Gamma be a closed, cocompact subgroup of a simply connected, solva ble Lie group G, such that Ad(G) Gamma has the same Zariski closure as Ad G. If alpha: Gamma --> GL(n)(R) is any finite-dimensional represen tation of Gamma, we show that alpha virtually extends to a representat ion of G. (By combining this with work of Margulis on lattices in semi simple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that i f Gamma is isomorphic to a closed, cocompact subgroup Gamma' of anothe r simply connected, solvable Lie group G', then any isomorphism from G amma to Gamma' extends to a crossed isomorphism from G to G'. In the s ame vein, we prove a more concrete form of Mostow's theorem that compa ct solvmanifolds with isomorphic fundamental groups are diffeomorphic.