Ya'acov Peterzil

Groups, measures, and the NIP

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s conjectures relating definably compact groups G in saturated o-minimal structures to compact Lie groups. We also prove some other structural results about such G, for example the existence of a left invariant finitely additive probability measure on definable subsets of G. We finally introduce a new notion “compact domination” (domination of a definable set by a compact space) and raise some new conjectures in the o-minimal case.

Definable Compactness and Definable Subgroups of o-Minimal Groups

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

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 .

Linear o-minimal structures

AbstractA linearly ordered structure

Simple algebraic and semialgebraic groups over real closed fields

\mathcal{M} = (M,< , \cdot \cdot \cdot )

THE CENTRAL PATH IN SMOOTH CONVEX SEMIDEFINITE PROGRAMS

is called o-minimal if every definable subset ofM is a finite union of points and intervals. Such an