{
  "id": "2209.12167",
  "version": "v1",
  "published": "2022-09-25T07:09:10.000Z",
  "updated": "2022-09-25T07:09:10.000Z",
  "title": "On equivalence relations induced by locally compact abelian Polish groups",
  "authors": [
    "Longyun Ding",
    "Yang Zheng"
  ],
  "comment": "14 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "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 $G$. The connected component of the identity of a Polish group $G$ is denoted by $G_0$. Let $G,H$ be locally compact abelian Polish groups. If $E(G)\\leq_B E(H)$, then there is a continuous homomorphism $S:G_0\\rightarrow H_0$ such that $\\ker(S)$ is non-archimedean. The converse is also true when $G$ is connected and compact. For $n\\in{\\mathbb N}^+$, the partially ordered set $P(\\omega)/\\mbox{Fin}$ can be embedded into Borel equivalence relations between $E({\\mathbb R}^n)$ and $E({\\mathbb T}^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-25T07:09:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15"
    ],
    "keywords": [
      "locally compact abelian polish groups",
      "right coset equivalence relation",
      "borel equivalence relations",
      "convergent sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}