{
  "id": "2406.01237",
  "version": "v1",
  "published": "2024-06-03T11:56:02.000Z",
  "updated": "2024-06-03T11:56:02.000Z",
  "title": "Charactarisation of distal actions of automorphisms on the space of one-parameter subgroups of Lie groups",
  "authors": [
    "Debamita Chatterjee",
    "Riddhi Shah"
  ],
  "categories": [
    "math.DS",
    "math.GR"
  ],
  "abstract": "For a connected Lie group $G$ and an automorphism $T$ of $G$, we consider the action of $T$ on Sub$_G$, the compact space of closed subgroups of $G$ endowed with the Chabauty topology. We study the action of $T$ on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of $G$. We relate the distality of the $T$-action on Sub$^p_G$ with that of the $T$-action on $G$ and characterise the same in terms of compactness of the closed subgroup generated by $T$ in Aut$(G)$ when $T$ acts distally on the maximal central torus and $G$ is not a vector group. We extend these results to the action of subgroups ${\\mathcal H}$ of Aut$(G)$ and equate the distal action of any closed subgroup ${\\mathcal H}$ on Sub$^p_G$ with that of every element in ${\\mathcal H}$. Moreover, we show that a connected Lie group $G$ acts distally on Sub$^p_G$ by conjugation if and only if $G$ is either compact or it is isomorphic to a direct product of a compact group and a vector group. Our results generalise some results proven by Shah and Yadav.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-03T11:56:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distal action",
      "automorphism",
      "closed subgroup",
      "connected lie group",
      "charactarisation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}