{
  "id": "2204.04594",
  "version": "v1",
  "published": "2022-04-10T03:55:59.000Z",
  "updated": "2022-04-10T03:55:59.000Z",
  "title": "On equivalence relations induced by Polish groups",
  "authors": [
    "Longyun Ding",
    "Yang Zheng"
  ],
  "comment": "41 pages, submitted",
  "categories": [
    "math.LO"
  ],
  "abstract": "The motivation of this article is to introduce a kind of orbit equivalence relations which can well describe structures and properties of Polish groups from the perspective of Borel reducibility. Given a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^\\omega/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. Let $G$ be a Polish group. (1) $G$ is a discrete countable group containing at least two elements iff $E(G)\\sim_BE_0$; (2) if $G$ is TSI uncountable non-archimedean, then $E(G)\\sim_BE_0^\\omega$; (3) $G$ is non-archimedean iff $E(G)\\le_B=^+$; (4) let $G$ be a non-CLI Polish group and $H$ a CLI Polish group, then $E(G)\\not\\le_BE_H^Y$ for any Polish $H$-space $Y$; (5) if $H$ is a non-archimedean Polish group but $G$ is not, then $E(G)\\not\\le_BE_H^Y$ for any Polish $H$-space $Y$. The notion of $\\alpha$-unbalanced Polish group for $\\alpha<\\omega_1$ is introduced. Let $G,H$ be Polish groups, $0<\\alpha<\\omega_1$. If $G$ is $\\alpha$-unbalanced but $H$ is not, then $E(G)\\not\\le_B E(H)$. For any Lie group $G$, denote $G_0$ the connected component of the identity element $1_G$. Let $G$ and $H$ be two separable TSI Lie groups. If $E(G)\\le_BE(H)$, then there exists a continuous locally injection $S:G_0\\to H_0$. Moreover, if $G_0,H_0$ are abelian, $S$ is a group homomorphism. Particularly, for $c_0,e_0,c_1,e_1\\in{\\mathbb N}$, $E({\\mathbb R}^{c_0}\\times{\\mathbb T}^{e_0})\\le_BE({\\mathbb R}^{c_1}\\times{\\mathbb T}^{e_1})$ iff $e_0\\le e_1$ and $c_0+e_0\\le c_1+e_1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-10T03:55:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "22A05",
      "22E15"
    ],
    "keywords": [
      "right coset equivalence relation",
      "orbit equivalence relations",
      "separable tsi lie groups",
      "non-archimedean polish group",
      "identity element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}