Some results on NIP groups and their Ellis groups

This paper has several parts. We begin by developing a theory of `piecewise (strong) f-genericity' in NIP groups, where we call a definable set piecewise (strong) f-generic if some union of finitely many translates of it is (strong) f-generic. We show that, in an NIP group, the definable sets that are not piecewise (strong) f-generic form an ideal. Our hope is that the corresponding piecewise (strong) f-generic types can provide a substitute in arbitrary NIP groups for the (strong) f-generic types of definably amenable NIP groups, and in the rest of the paper we give several applications.
Two of the applications deal with the Ellis group of an NIP group. Let $T$ be an NIP theory, $G$ a definable group, and $M$ a model. In our first result we show that the size of the Ellis group of $G(M)$ is bounded above by $2^{|T|}$, independent of the choice of $M$, giving a substantial step towards the question of whether the isomorphism type is independent of $M$. In our second result, inspired by a theorem of Hrushovski, we show that, if $T$ and $M$ are countable and the formulas of $T$ have uniformly bounded VC-codensity, then the Ellis group of $G(M)$ has `finite Archimedean rank', ie its connected component is profinite-by-Lie. A crucial tool for us in both results is the recent result of Chernikov-Gannon-Krupiński and Basso-Zucker that the $\tau$-topology on the Ellis group is Hausdorff.
Finally, we use our techniques to obtain a `local' result valid in arbitrary NIP theories, without the assumption of uniformly bounded VC-codensity: for any `bi-invariant' formula $\phi(x,y)$, the group $G/G^{00}_\phi$ has finite Archimedean rank. More precisely, if the VC-codensity of $\phi(x,y)$ is at most $\delta$, then $G/G^{00}_\phi$ is an inverse limit of compact Lie groups of dimension at most $(4\delta)^2$. This connects to, though is different than, a question of Hrushovski's.