turkmath.org

Among the problems of the model theory of a group $G$ is the one of knowing which subgroups of $G$ are definable by a formula in the pure language of groups. I will focus mainly on the class of NIP groups. A group $G$ does Not have the Independence Property (or is NIP for short) if every formula $f(x; y_1,...,y_n)$ defines a family of subsets of $G$ having a finite VC-dimension (the so called famous Vapnik-Chervonenkis notion of dimension coming from statistics). The class of NIP groups include linear algebraic groups over the field of real numbers, over separably closed fields and over the field of $p$-adic numbers.