Archimedean Superrigidity of SolvableS-Arithmetic Groups Original Research Article

Let image be a connected, solvable linear algebraic group over a number fieldK, letSbe a finite set of places ofKthat contains all the infinite places, and let image(S) be the ring ofS-integers ofK. We define a certain closed subgroup[formula]of imageS = ∏v set membership, variant SimageKvthat contains imageimage(S), and prove that imageimage(S)is a superrigid lattice in[formula], by which we mean that finite-dimensional representations α : imageimage(S) → GLn(image) more or less extend to representations of[formula]. The subgroup[formula]may be a proper subgroup of imageSfor only two reasons. First, it is well known that imageimage(S)is not a lattice in imageSif image has nontrivialK-characters, so one passes to a certain subgroup imageS(1). Second, imageimage(S)may fail to be Zariski dense in imageS(1)in an appropriate sense; in this sense, the subgroup[formula]is the Zariski closure of imageimage(S)in imageS(1). Furthermore, we note that a superrigidity theorem for many nonsolvableS-arithmetic groups can be proved by combining our main theorem with the Margulis Superrigidity Theorem.