Let T be a closed, cocompact subgroup of a simply connected, solvable Lie group G, such that Ad_g T has the same Zariski closure as AdG. If a:T-GL_n(R) is any finite-dimensional representation of T, we show that a virtually extends to a representation of G. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if T is isomorphic to a closed, cocompact subgroup T' of another simply connected, oslvable Lie group G', then any isomorphism from T to T' extends to a crossed isomorphism from G to G'. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvemnifolds with isomorphic fundamental groups are diffeomorphic.
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