{
  "id": "1912.01446",
  "version": "v1",
  "published": "2019-12-03T15:07:43.000Z",
  "updated": "2019-12-03T15:07:43.000Z",
  "title": "Distal Actions of Automorphisms of Nilpotent Groups $G$ on Sub_$G$ and Applications to Lattices in Lie Groups",
  "authors": [
    "Rajdip Palit",
    "Riddhi Shah"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "For a locally compact group $G$, we study the distality of the action of automorphisms $T$ of $G$ on ${\\rm Sub}_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. For a certain class of discrete groups $G$, we show that $T$ acts distally on ${\\rm Sub}_G$ if and only if $T^n$ is the identity map for some $n\\in{\\mathbb N}$. As an application, we get that for a $T$-invariant lattice $\\Gamma$ in a simply connected nilpotent Lie group $G$, $T$ acts distally on ${\\rm Sub}_G$ if and only if it acts distally on ${\\rm Sub}_\\Gamma$. This also holds for any closed $T$-invariant co-compact subgroup $\\Gamma$. For a lattice $\\Gamma$ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on ${\\rm Sub}_\\Gamma$. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group from that in a nilpotent Lie group. For torsion-free compactly generated nilpotent (metrizable) groups $G$, we obtain the following characterisation: $T$ acts distally on ${\\rm Sub}_G$ if and only if $T$ is contained in a compact subgroup of ${\\rm Aut}(G)$. Using these results, we characterise the class of such groups $G$ which act distally on ${\\rm Sub}_G$. We also show that any compactly generated distal group $G$ is Lie projective. As a consequence, we get some results on the structure of compactly generated nilpotent groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-03T15:07:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nilpotent groups",
      "distal actions",
      "automorphisms",
      "connected nilpotent lie group",
      "connected solvable lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}