Generalised 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 (from [9] for example). 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] generalised 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 generalised 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; (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 SG(M) are definable, one can just work with the definable (instead of externally definable) objects in the above results.