Leeds

Newelski suggested that topological dynamics could be used to extend "stable group theory" results outside the stable context. Given a group G, it acts on the left on its type space S_G(M), i.e. (G,S_G(M)) is a G-flow. If every type is definable, S_G(M) can be equipped with a semigroup structure *, and it is isomorphic to the enveloping Ellis semigroup of the flow. The topological dynamics of (G,S_G(M)) is coded in the Ellis semigroup and in its minimal G-invariant subflows, which coincide with the left ideals I of S_G(M). Such ideals contain at least an idempotent r, and r*I forms a group, called "ideal group". Newelski proved that in stable theories and in o-minimal theories r*I is abstractly isomorphic to G/G^{00} as a group. He then asked if this happens for any NIP theory. Pillay recently extended the result to fsg groups; we found instead a counterexample to Newelski`s conjecture in SL(2,R), for which G/G^{00} is trivial but we show r*I has two elements. This is joint work with Jakub Gismatullin and Anand Pillay.

After Sela and Kharlampovich-Myasnikov independently proved that non abelian free groups share the same common theory model theoretic interest for the subject arose.

 In this talk I will present a survey of results around this theory starting with basic model theoretic properties mostly coming from the connectedness of the free group (Poizat).

Then I will sketch our proof with C.Perin for the homogeneity of non abelian free groups and I will give several applications, the most important being the description of forking independence.

 In the last part I will discuss a list of open problems, that fit in the context of geometric stability theory, together with some ideas/partial answers to them.