{
  "id": "1909.04397",
  "version": "v1",
  "published": "2019-09-10T10:42:40.000Z",
  "updated": "2019-09-10T10:42:40.000Z",
  "title": "Distal Actions of Automorphisms of Lie Groups $G$ on $\\rm Sub_{G}$",
  "authors": [
    "Riddhi Shah",
    "Alok Kumar Yadav"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "For a locally compact metrizable group $G$, we study the action of $\\rm Aut(G)$ on $\\rm Sub_G$, the set of closed subgroups of $G$ endowed with the Chabauty topology. Given an automorphism $T$ of $G$, we relate the distality of the $T$-action on $\\rm Sub_G$ with that of the $T$-action on $G$ under a certain condition. If $G$ is a connected Lie group, we characterise the distality of the $T$-action on $\\rm Sub_G$ in terms of compactness of the closed group generated by $T$ in $\\rm Aut(G)$ under certain conditions on the center of $G$ or on $T$ as follows: $G$ has no compact central subgroup of positive dimension or $T$ is unipotent or $T$ is contained in the connected component of the identity in $\\rm Aut(G)$. Moreover, we also show that a connected Lie group $G$ acts distally on $\\rm Sub_G$ if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. All the results on the Lie groups mentioned above hold for the action on $\\rm Sub^a_G$, a subset of $\\rm Sub_G$ consisting of closed abelian subgroups of $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-10T10:42:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distal actions",
      "automorphism",
      "connected lie group",
      "compact central subgroup",
      "chabauty topology"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}