Definably Simple Groups in O-minimal Structures

Let G = 〈G, ·〉 be a group definable in an o-minimal structure M. A subset H of G is G-definable if H is definable in the structure 〈G, ·〉 (while definable means definable in the structure M). Assume G has no Gdefinable proper subgroup of finite index. In this paper we prove that if G has no nontrivial abelian normal subgroup, then G is the direct product of G-definable subgroups H1, . . . ,Hk such that each Hi is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture. This is the first of two papers around groups definable in o-minimal structures and semialgebraic groups over real closed fields. An o-minimal structure is a structureM = 〈M,<, ....〉 where < is a dense linear ordering of M , and any definable subset of M is a finite union of intervals (with endpoints in M∪{±∞}) and points. A group G is said to be definable inM if both G and the graph of the group operation on G are definable sets inM (i.e. definable subsets of M, M for some n). The typical example is G = H(R) where H is an algebraic group defined over a real closed field R. (Take M = 〈R, <,+, ·, 〉.) We show a converse: suppose that G is definable in some o-minimal structure and that G is nonabelian and has no proper nontrivial normal subgroup definable in the structure 〈G, ·〉 (we say that G is G-definably simple). Then G is isomorphic to an (open) semialgebraic subgroup of finite index of a group of the form H(R), where R is a real closed field and H is an R-simple algebraic group. This gives a positive answer to the o-minimal analogue of the (yet unproved) Cherlin-Zilber conjecture: any simple group of finite Morley rank is an algebraic group over an algebraically closed field. The strategy of our proof is closely related to Poizat’s approach ([12]) to Cherlin’s conjecture. Given G definable in o-minimalM, we try to find a real closed field R definable inM which is intimately connected to G. We then try to show that G is definably (inM) isomorphic to a linear semialgebraic group over R. The first step is made possible by, among other things, the Trichotomy theorem. The second step goes through developing Lie theory over o-minimal expansions of real closed fields. This second step is possible, because, once we have a real closed field R definable in an o-minimal structureM, then definable (inM) functions on R are piecewise as differentiable as one wants. In practice it is convenient to work with centerless and “semisimple” groups, namely groups with no nontrivial normal abelian subgroups, and for these we prove Received by the editors February 25, 1998. 2000 Mathematics Subject Classification. Primary 03C64, 22E15, 20G20; Secondary 12J15. The second and the third authors were partially supported by NSF. c ©2000 American Mathematical Society