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

Constructing Simplicial Branched Covers

Branched covers are applied frequently in topology - most prominently in the construction of closed oriented PL d-manifolds. In particular, strong bounds for the number of sheets and the topology of the branching set are known for dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain a simplicial covering space (the partial unfolding) of a given simplicial complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255, 2003]. We present a large class of branched covers which can be constructed via the partial unfolding. In particular, for d<=4 every closed oriented PL d-manifold is the partial unfolding of some polytopal d-sphere.Comment: 15 pages, 8 figures, typos corrected and conjecture adde