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. 1 Introduction and

Abstract. We consider groups definable in an o-minimal expansion of a real closed field. To each definable group G is associated in a canonical way a real Lie group G/G 00 which, in the definably compact case, captures many of the algebraic and topological features of G. In particular, if G is definably compact and definably connected, the definable fundamental group of G is isomorphic to the fundamental group of G/G 00. However the functorial properties of the isomorphism have so far not been investigated. Moreover from the known proofs it is not easy to understand what is the image under the isomorphism of a given generator. Here we clarify the situation using the “compact domination conjecture ” proved by Hrushovski, Peterzil and Pillay. We construct a natural homomorphism between the definable fundamental groupoid of G and the fundamental groupoid of G/G 00 which is equivariant under the action of G and induces a natural isomorphism on the fundamental groups. We use this to prove the following result. Let G and G ′ be two definably compact definably connected groups with isomorphic associated Lie groups. Then G and G ′ are definably homotopy equivalent. Moreover given a finite subgroup Γ of G, there is a definable homotopy equivalence f: G → G ′ that restricted to Γ is an isomorphism onto its image and such that f(cx) = f(c)f(x) for all c ∈ Γ and x ∈ G. In the semisimple case a stronger result holds: any Lie isomorphism from G/G 00 to G ′ /G ′00 induces a definable isomorphism from G to G ′. 1.

ABSTRACT. We introduce the o-minimal LS-category of definable sets in o-minimal expansions of ordered fields and we establish a relation with the semialgebraic and the classical one. We also study the o-minimal LScategory of definable groups. Along the way, we show that two definably connected definably compact definable groups G and H are definable homotopy equivalent if and only if L(G) and L(H) are homotopy equivalent, where L is the functor which associates to each definable group its corresponding Lie group via Pillay’s conjecture. 1

We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.

We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers. With partial support from the FCT (Fundação para a Ciência e Tecnologia) program