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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 $α\colon \Ga_{\theints} \to \GL_n(\real)$ 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.