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Archimedean Superrigidity of SolvableS-Arithmetic Groups

✍ Scribed by Dave Witte

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
235 KB
Volume
187
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let ‫އ‬ be a connected, solvable linear algebraic group over a number field K, let Ž . S be a finite set of places of K that contains all the infinite places, and let O O S be the ring of S-integers of K. We define a certain closed subgroup ‫އ‬ of

‫އ‬ s Ł ‫އ‬ that contains ‫އ‬ , and prove that ‫އ‬ is a superrigid lattice in

‫އ‬

, by which we mean that finite-dimensional representations

Ž . GL ‫ޒ‬ more or less extend to representations of ‫އ‬ . The subgroup ‫އ‬ may

be a proper subgroup of ‫އ‬ for only two reasons. First, it is well known that ‫އ‬

is not a lattice in ‫އ‬ if ‫އ‬ has nontrivial K-characters, so one passes to a certain S subgroup ‫އ‬ Ž1. . Second, ‫އ‬ may fail to be Zariski dense in ‫އ‬ Ž1. in an appropriate

sense; in this sense, the subgroup ‫އ‬ is the Zariski closure of ‫އ‬ in ‫އ‬ .

Furthermore, we note that a superrigidity theorem for many nonsolvable S-arithmetic groups can be proved by combining our main theorem with the Margulis Superrigidity Theorem.

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