Sergei Starchenko

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

Definable homomorphisms of abelian groups in o-minimal structures

Abstract We investigate the group H of definable homomorphisms between two definable abelian groups A and B , in an o-minimal structure N . We prove the existence of a “large”, definable subgroup of H . If H contains an infinite definable set of homomorphisms then some definable subgroup of B (equivalently, a definable quotient of A ) admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure N but also in any structure definable in N .

Expansions of algebraically closed fields in o-minimal structures

Abstract. We develop a notion of differentiability over an algebraically closed field K of characteristic zero with respect to a maximal real closed subfield R. We work in the context of an o-minimal expansion

Simple algebraic and semialgebraic groups over real closed fields

\cal {R}

A growth dichotomy for o-minimal expansions of ordered groups

of the field R and obtain many of the standard results in complex analysis in this setting. In doing so we use the topological approach to complex analysis developed by Whyburn and others. We then prove a model theoretic theorem that states that the field R is definable in every proper expansion of the field K all of whose atomic relations are definable in

Definability of restricted theta functions and families of abelian varieties

\cal {R}

Tame Complex Analysis and o-minimality

. One corollary of this result is the classical theorem of Chow on projective analytic sets.

Model Theory with Applications to Algebra and Analysis: Complex analytic geometry in a nonstandard setting

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 compositionisomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

Vapnik–Chervonenkis Density in Some Theories without the Independence Property, II

Let R be an o-minimal expansion of a divisible ordered abelian group (R, <, +, 0, 1) with a distinguished positive element 1. Then the following dichotomy holds: Either there is a 0-definable binary operation · such that (R, <, +, ·, 0, 1) is an ordered real closed field; or, for every definable function f : R → R there exists a 0-definable λ ∈ {0} ∪ Aut(R, +) with limx→+∞[f(x) − λ(x)] ∈ R. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure M := (M, <, . . . ) there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) M-definable groups with underlying set M . R. Poston showed in [8] that given an o-minimal expansion R of (R, <, +), if multiplication is not definable in R, then for every definable function f : R → R there exist r, c ∈ R such that limx→+∞[f(x) − rx] = c. In this paper, this fact is generalized appropriately for o-minimal expansions of arbitrary ordered groups. We say that an expansion (G, <, ∗, . . . ) of an ordered group (G, <, ∗) is linearly bounded (with respect to ∗) if for each definable function f : G → G there exists a definable λ ∈ End(G, ∗) such that ultimately |f(x)| ≤ λ(x). (Here and throughout, ultimately abbreviates “for all sufficiently large positive arguments”.) We now list the main results of this paper. Let R := (R, <, . . . ) be o-minimal. Theorem A (Growth Dichotomy). Suppose that R is an expansion of an ordered group (R, <, +). Then exactly one of the following holds: (a) R is linearly bounded; (b) R defines a binary operation · such that (R, <, +, ·) is an ordered real closed field. If R is linearly bounded, then for every definable f : R → R there exist c ∈ R and a definable λ ∈ {0} ∪ Aut(R, +) with limx→+∞[f(x)− λ(x)] = c. Theorem B. Suppose that R is a linearly bounded expansion of an ordered group (R, <, +, 0, 1) with 1 > 0. Then every definable endomorphism of (R, +) is 0-definable. If R′ (with underlying set R′) is elementarily equivalent to R, then the ordered division ring of all R′-definable endomorphisms of (R′, +) is canonically isomorphic to the ordered division ring of all R-definable endomorphisms of (R, +). The growth dichotomy imposes some surprising constraints on continuous definable groups with underlying set R. (Here and throughout, all topological notions are taken with respect to the product topologies induced by the order topology.) Received by the editors June 5, 1996. 1991 Mathematics Subject Classification. Primary 03C99; Secondary 06F20, 12J15, 12L12. The first author was supported by NSF Postdoctoral Fellowship No. DMS-9407549. c ©1998 American Mathematical Society

Computing o-minimal topological invariants using differential topology

We consider some classical maps from the theory of abelian varieties and their moduli spaces and prove their definability, on restricted domains, in the o-minimal structure