{
  "id": "2010.12958",
  "version": "v1",
  "published": "2020-10-24T19:15:25.000Z",
  "updated": "2020-10-24T19:15:25.000Z",
  "title": "Dynamics of Actions of Automorphisms of Discrete Groups $G$ on Sub$_G$ and Applications to Lattices in Lie Groups",
  "authors": [
    "Rajdip Palit",
    "Manoj B. Prajapati",
    "Riddhi Shah"
  ],
  "comment": "36 pages",
  "categories": [
    "math.GR",
    "math.DS"
  ],
  "abstract": "For a discrete group $G$ and the compact space Sub$_G$ of (closed) subgroups of $G$ endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of $G$ on Sub$_G$ in terms of distality and expansivity. We also study the structure and properties of lattices $\\Gamma$ in a connected Lie group. In particular, we show that the unique maximal solvable normal subgroup of $\\Gamma$ is polycyclic and the corresponding quotient of $\\Gamma$ is either finite or admits a cofinite subgroup which is a lattice in a connected semisimple Lie group with certain properties. We also show that Sub$^c_\\Gamma$, the set of cyclic subgroups of $\\Gamma$, is closed in Sub$_\\Gamma$. We prove that an infinite discrete group $\\Gamma$ which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on Sub$^c_\\Gamma$, while only the finite order automorphisms of $\\Gamma$ act distally on Sub$^c_\\Gamma$. For an automorphism $T$ of a connected Lie group $G$ and a $T$-invariant lattice $\\Gamma$ in $G$, we compare the behaviour of the actions of $T$ on Sub$_G$ and Sub$_\\Gamma$ in terms of distality. We put certain conditions on the structure of the Lie group $G$ under which we show that $T$ acts distally on Sub$_G$ if and only if it acts distally on Sub$_\\Gamma$. We construct counter examples to show that this does not hold in general if the conditions on the Lie group are relaxed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-24T19:15:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B05",
      "22E40"
    ],
    "keywords": [
      "automorphism",
      "connected lie group",
      "unique maximal solvable normal subgroup",
      "applications",
      "connected semisimple lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}