Atticus Stonestrom - Approximate groups in NIP structures
112
ICJ
Recall that a subset X of a group is said to be an approximate group if the product set XX:=\{ab:a,b\in X\} can be covered by finitely many translates of X. I will discuss a work-in-progress which attempts to give a structure theory for approximate groups definable in NIP theories. First I will present an example of an approximate group definable in the universal cover of SL(2,R) that has no "locally compact model"; this gives a NIP counterexample to a conjecture of Massicot-Wagner, first disproved by Hrushovski, Krupiński, and Pillay. I will then discuss some tentative positive structural results, some of which are of interest even in the case of definable groups. Among other things I will discuss the shape of Hrushovski’s “generalized locally compact models” in the NIP setting.