Simple algebraic and semialgebraic groups over real closed fields

Ya'acov Peterzil, Anand Pillay, Sergei Starchenko

Research output: Contribution to journalArticlepeer-review


We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. \Ve discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

Original languageEnglish
Pages (from-to)4421-4450
Number of pages30
JournalTransactions of the American Mathematical Society
Issue number10
StatePublished - 2000

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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