1996/11/19 by Dave Witte, Witte, Dave
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT
paper · pdf · doi:10.48550/arxiv.math/9611219
arxiv created 1996/11/19 · openalex publication_date 1996/11/19 · arxiv updated 2016/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \GaS = ∏v ∈ S \GaKv 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 → \GLn(\real) more-or-less extend to representations of~\GOS. The subgroup~\GOS may be a proper subgroup of~\GaS for only two reasons. First, it is well known that \Ga\theints is not a lattice in~\GaS 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.