{
  "id": "2406.06082",
  "version": "v1",
  "published": "2024-06-10T08:03:37.000Z",
  "updated": "2024-06-10T08:03:37.000Z",
  "title": "The class and dynamics of $α$-balanced Polish groups",
  "authors": [
    "Shaun Allison",
    "Aristotelis Panagiotopoulos"
  ],
  "categories": [
    "math.LO",
    "math.DS",
    "math.GN",
    "math.GR"
  ],
  "abstract": "For each ordinal $\\alpha<\\omega_1$, we introduce the class of $\\alpha$-balanced Polish groups. These classes form a hierarchy that completely stratifies the space between the class of Polish groups admitting a two-side-invariant metric (TSI) and the class of Polish groups admitting a complete left-invariant metric (CLI). We establish various closure properties, provide connections to model theory, and we develop a boundedness principle for CLI groups by showing that $\\alpha$-balancedness is an initial segment of a regular coanalytic rank. In the spirit of Hjorth's turbulence theory we also introduce \"generic $\\alpha$-unbalancedness\": a new dynamical condition for Polish $G$-spaces which serves as an obstruction to classification by actions of $\\alpha$-balanced Polish groups. We use this to provide, for each $\\alpha<\\omega_1$, an action of an $\\alpha$-balanced Polish group whose orbit equivalence relation is strongly generically ergodic against actions of any $\\beta$-balanced Polish group with $\\beta<\\alpha$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-10T08:03:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "54H05",
      "37B02",
      "54H11",
      "22A05"
    ],
    "keywords": [
      "balanced polish group",
      "hjorths turbulence theory",
      "orbit equivalence relation",
      "polish groups admitting",
      "complete left-invariant metric"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}