2016/12/05 by Krzysztof Krupiński, KRZYSZTOF KRUPIŃSKI, Anand Pillay +1 · 3 citations
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Advanced Operator Algebra Research
paper · doi:10.1017/s0305004116000967
Abstract 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 S G ( M ) are definable, one can just work with the definable (instead of externally definable) objects in the above results.