Submission date: 30 June 2014.

Abstract:

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. 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 (definable) generalized Bohr compactfication of G; (definable) strong amenability. Among other things, we essentially prove: (i) The “new” invariant G^*/(G^*)^{000}_M lies in between the (definable) generalized Bohr compactification and the (definable) Bohr compactification, and these all coincide when G is (definably) strongly amenable, (ii) the quotient of the (definable) Bohr compactification of G by G^*/(G^*)^{000}_M 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 definably amenable iff it is definably strongly amenable.

Mathematics Subject Classification: 03C45, 54H20, 37B05, 20A15

Keywords and phrases: [externally] definable Bohr compactification, model-theoretic connected components, definable strong amenability

Full text arXiv 1406.7730: pdf, ps.