{
  "id": "1612.07560",
  "version": "v1",
  "published": "2016-12-22T12:03:48.000Z",
  "updated": "2016-12-22T12:03:48.000Z",
  "title": "Amenability, definable groups, and automorphism groups",
  "authors": [
    "Krzysztof Krupinski",
    "Anand Pillay"
  ],
  "categories": [
    "math.LO",
    "math.DS",
    "math.GN"
  ],
  "abstract": "We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, if $M$ is a countable, $\\omega$-categorical structure and $Aut(M)$ is amenable, as a topological group, then the Lascar Galois group $Gal_{L}(T)$ of the theory $T$ of $M$ is compact, Hausdorff (also over any finite set of parameters), that is $T$ is G-compact. An essentially special case is that if $Aut(M)$ is extremely amenable, then $Gal_{L}(T)$ is trivial, so, by a theorem of Lascar, the theory $T$ can be recovered from its category $Mod(T)$ of models. On the side of definable groups, we prove for example that if $G$ is definable in a model $M$, and $G$ is definably amenable, then the connected components ${G^{*}}^{00}_{M}$ and ${G^{*}}^{000}_{M}$ coincide, answering positively a question from an earlier paper of the authors. We also take the opportunity to further develop the model-theoretic approach to topological dynamics, obtaining for example some new invariants for topological groups, as well as allowing a uniform approach to the theorems above and the various categories.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-22T12:03:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03C45",
      "54H20",
      "54H11",
      "43A07"
    ],
    "keywords": [
      "automorphism group",
      "definable groups",
      "lascar galois group",
      "connected components",
      "topological group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}