Generalized Bohr compactification and model-theoretic connected components

For a group $G$ first order definable in a structure $M$, we continue the study of the definable topological dynamics of $G$. The special case when all subsets of $G$ are definable in the given structure $M$ is simply the usual topological dynamics of the discrete group $G$; in particular, in this case, the words externally definable and definable can be removed in the results described below. Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant $G^{*}/(G^{*})^{000}_{M}$ of $G$, which appears to be new in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalized Bohr compactification of $G$; [externally definable] strong amenability. Among other things, we essentially prove: (i) The new invariant $G^{*}/(G^{*})^{000}_{M}$ lies in between the externally definable generalized Bohr compactification and the definable Bohr compactification, and these all coincide when $G$ is definably strongly amenable and all types in $S_G(M)$ are definable, (ii) the kernel of the surjective homomorphism from $G^*/(G^*)^{000}_M$ to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup, and (iii) when $Th(M)$ is NIP, then $G$ is [externally] definably amenable iff it is externally definably strongly amenable. In the situation when all types in $S_G(M)$ are definable, one can just work with the definable (instead of externally definable) objects in the above results.