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Archimedean superrigidity of solvable S-arithmetic groups
Let \Ga be a connected, solvable linear algebraic group over a number
field~, let be a finite set of places of~ that contains all the
infinite places, and let \theints be the ring of -integers of~. 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 -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
-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 is a semisimple Levi subgroup of a connected Lie group~,
is a Borel -space with finite invariant measure, and \alpha \colon X \times
G \to \GL_n(\real) is a Borel cocycle. Assume has finite center, and that
the real rank of every simple factor of~ is at least two. We show that if
is ergodic on~, and the restriction of~ to~ is
cohomologous to a homomorphism (modulo a compact group), then, after passing to
a finite cover of~, the cocycle itself is cohomologous to a
homomorphism (modulo a compact group)
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