Anand Pillay

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.

On groups and fields definable in o-minimal structures

The structure M = (M, <, R1, R2,…) is o-minimal if every definable set X ⊂ M is a finite union of intervals (a, b) and points. Let G be a group definable in M (i.e. G is a definable subset of Mn and the graph of multiplication is also definable). We show that G can be definably equipped with the structure of a ‘manifold’ over M in which multiplication and inversion are continuous. In the special case M = (R, <, +, ·) our construction gives G the structure of a Nash group. These results are also used to show that an infinite field definable in an o-minimal structure is real closed or algebraically closed.

Definable sets in ordered structures

On introduit la notion de theorie O-minimale des structures ordonnees, une telle theorie etant telle que les sous-ensembles definissables de ses modeles soient particulierement simples

On NIP and invariant measures

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd(A), (ii) analogous statements for Keisler measures and definable groups, including the fact that G 000 = G 00 for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o-minimal expansions of real closed fields.

Generic structures and simple theories

Abstract We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field (possibly imperfect) is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

Weakly Normal Groups

Publisher Summary This chapter describes that a group G is weakly normal if and only if every definable X ⊂ G n is a Boolean combination of cosets of acl (ϕ)-definable subgroups ∀n if and only if every definable X ⊂ G n is a Boolean combination of cosets of definable subgroups (of G n ) ∀n. The chapter describes some equivalent conditions to that of weak normality and some basic results on stable groups. A locally connected definable subgroup of a weakly normal group is acl (ϕ)-definable. The chapter shows that if G is a weakly normal group, any p ∊ S(G) is determined by the definable cosets in p. The above result is enough to prove equivalence in the theorem, “an Abelian structure is an Abelian group A, together with distinguished subgroups of A n for various ns. Any Abelian structure is interpretable in a module.”

Coordinatisation and Canonical Bases in Simple Theories

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a / E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated model M of a complete theory T . All types, sets and sequences will have size smaller than the size of M . We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

GROUPS DEFINABLE IN LOCAL FIELDS AND PSEUDO-FINITE FIELDS

Using model-theoretic methods we prove:Theorem AIf G is a Nash group over the real or p-adic field, then there is a Nash isomorphism between neighbourhoods of the identity of G and of the set of F-rational points of an algebraic group defined over F.Theorem BLet G be a connected affine Nash group over ℝ. Then G is Nash isogeneous with the (real) connected component of the set of real points of an algebraic group defined over ℝ.Theorem CLet G be a group definable in a pseudo-finite field F. Then G is definably virtually isogeneous with the set of F-rational points of an algebraic group defined over F.

Hyperimaginaries and Automorphism Groups

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [?], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M). In section 2 we show that if T is simple and canonical bases of Lascar strong types exist in M eq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In section ∗Partially supported by an NSF grant This work was begun during a visit of the two authors to the Centre de Recerca Matemèmatica, Institut d’Estudis Catalans. The authors wish to express their gratitude for its support and hospitability

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