Archimedean superrigidity of solvable S-arithmetic groups

Let \Ga be a connected, solvable linear algebraic group over a number field~$K$, let $S$ be a finite set of places of~$K$ that contains all the infinite places, and let \theints be the ring of $S$-integers of~$K$. We define a certain closed subgroup~\GOS of \Ga_S = \prod_{v \in S} \Ga_{K_v} that contains \Ga_{\theints}, and prove that \Ga_{\theints} is a superrigid lattice in~\GOS, by which we mean that finite-dimensional representations \alpha\colon \Ga_{\theints} \to \GL_n(\real) more-or-less extend to representations of~\GOS. The subgroup~\GOS may be a proper subgroup of~\Ga_S for only two reasons. First, it is well known that \Ga_{\theints} is not a lattice in~\Ga_S if \Ga has nontrivial $K$-characters, so one passes to a certain subgroup \GS. Second, \Ga_{\theints} may fail to be Zariski dense in \GS in an appropriate sense; in this sense, the subgroup \GOS is the Zariski closure of~\Ga_{\theints} in~\GS. Furthermore, we note that a superrigidity theorem for many non-solvable $S$-arithmetic groups can be proved by combining our main theorem with the Margulis Superrigidity Theorem

Cocycle superrigidity for ergodic actions of non-semisimple Lie groups

Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and \alpha \colon X \times G \to \GL_n(\real) is a Borel cocycle. Assume $L$ has finite center, and that the real rank of every simple factor of~$L$ is at least two. We show that if $L$ is ergodic on~$X$, and the restriction of~$\alpha$ to~$X \times L$ is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of~$X$, the cocycle $\alpha$ itself is cohomologous to a homomorphism (modulo a compact group)