Linear groups definable in o-minimal structures

Abstract

We study subgroups G of GL(n, R) definable in o-minimal expansions M = (R, +,·,...) of a real closed field R. We prove several results such as: (a) G can be defined using just the field structure on R together with, if necessary, power functions, or an exponential function definable in M. (b) If G has no infinite, normal, definable abelian subgroup, then G is semialgebraic. We also characterize the definably simple groups definable in o-minimal structures as those groups elementarily equivalent to simple Lie groups, and we give a proof of the Kneser-Tits conjecture for real closed fields. © 2002 Elsevier Science.