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.

We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory. 2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

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.

We prove several structural results on definable, definably compact groups G in o-minimal expansions of real closed fields such as (i) G is definably an almost direct product of a semisimple group and a commutative group, (ii) (G, ·) is elementarily equivalent to (G/G 00 , ·). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as deducing the full compact domination conjecture for definably compact groups from the semisimple and commutative cases which were already settled. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups.

Abstract. A subset X of a group G is called left generic if finitely many left translates of X cover G. Our main result is that if G is a definably compact group in an o-minimal structure and a definable X ⊆ G is not right generic then its complement is left generic.Among our additional results are (i) a new condition equivalent to definable compactness, (ii) the existence of a finitely additive invariant measure on definable sets in a definably compact group G in the case where G = * H for some compact Lie group H (generalizing results from [1]), and (iii) in a definably compact group every definable subsemi-group is a subgroup.Our main result uses recent work of Alf Dolich on forking in o-minimal stuctures.1. Introduction. In his paper [9], the second author formulated a conjecture on definably compact groups in (saturated) o-minimal structures. The conjecture says, roughly, that such a group G has a smallest typedefinable subgroup G 00 of bounded index, and that G/G 00 , when equipped with the "logic topology", is a compact Lie group whose dimension equals the o-minimal dimension of G. In [2] the existence of the smallest typedefinable subgroup of bounded index G 00 was established, as well as the fact that G/G 00 is a compact Lie group. However, it remains open whether G 00 is at all different from G and, in particular, if the Lie group G/G 00 and G have the same dimension. When restricted to commutative G, the second (as yet unproved) part of the conjecture is equivalent to "G 00 is torsion-free".One of the missing ingredients towards establishing the full conjecture seems to be a good understanding of "generic" definable sets in definably 2000 Mathematics Subject Classification: 03C64, 22E15.