Simplicity and the Lascar Group

This paper contains a series of easy constructions and observations relating to the Lascar group and to simple theories. 1 In x1 we review basic model theoretic ideas, relating mostly to model completion and saturated models. We do so in order to introduce a framework very slightly more general than the usual rst-order one that will be useful to us, and that we hope may be useful in the future. We will refer to this as "Robinson theories". In x2 we give an account of Lascar's beautiful construction, associating a compact topological group to rst-order theories. Our description, innuenced by work of Kim-Pillay, applies to all rst order theories. For "G-compact" theories the results coincides with the full Lascar group; for others, if there are any others, it gives a quotient of the full Lascar group. We present the Lascar group as an automorphism group of a compact topological structure associated naturally with the theory, that we call the Kim-Pillay space. In x3 we nd a connection between the Lascar group and certain spaces of theories. In particular, we see that a necessary condition for the existence of a theory with connected Lascar group (in a certain class of theories closed under interpretations), is that there exist a continuous path in the space of theories (within the given class), interpolating between the theory of the empty graph to the theory of the complete graph. It is worth noting that even if we start with a rst order theory, this analysis necessarily involves Robinson theories; it would not have been possible without the extension of the framework in x1. The Lascar group was brought into prominence in recent work of Kim and Pillay, on simple theories. We will discuss simplicity brieey in the introduction to x4. This property was introduced in Sh1] as a generalization of stability. After a decade of neglect, a few years of intense activity by a number of workers, sparked by Kim's thesis and by some work in nite rank, brought the state of knowledge to nearly the same level as for stable theories. It was found that a theory of independence can be developed that is as coherent and satisfactory as in the stable context, though necessarily with some diierent features. However, the general theory as described by Kim written with the support of the Miller Institute while visiting the