{
  "id": "1406.7730",
  "version": "v2",
  "published": "2014-06-30T13:35:30.000Z",
  "updated": "2015-03-29T08:55:12.000Z",
  "title": "Generalized Bohr compactification and model-theoretic connected components",
  "authors": [
    "Krzysztof Krupinski",
    "Anand Pillay"
  ],
  "categories": [
    "math.LO",
    "math.GN",
    "math.GR"
  ],
  "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 $G$; 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 compactfication 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-30T13:35:30.000Z",
      "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.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-29T08:55:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C45",
      "54H20",
      "37B05",
      "20A15"
    ],
    "keywords": [
      "generalized bohr compactification",
      "model-theoretic connected components",
      "topological dynamics",
      "compact hausdorff group",
      "dense normal subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.7730K"
    }
  }
}