{
  "id": "2106.04242",
  "version": "v1",
  "published": "2021-06-08T10:39:10.000Z",
  "updated": "2021-06-08T10:39:10.000Z",
  "title": "Twisted Conjugacy in Linear Algebraic Groups II",
  "authors": [
    "Sushil Bhunia",
    "Anirban Bose"
  ],
  "comment": "22 pages",
  "categories": [
    "math.GR"
  ],
  "abstract": "Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $Aut(G)$ the group of all algebraic group automorphisms of $G$. For every $\\varphi\\in Aut(G)$ let $\\mathcal{R}(\\varphi)$ denote the set of all orbits of the $\\varphi$-twisted conjugacy action of $G$ on itself (given by $(g,x)\\mapsto gx\\varphi(g^{-1})$, for all $g,x\\in G$). We say that $G$ satisfies the $R_\\infty$-property if $\\mathcal{R}(\\varphi)$ is infinite for every $\\varphi\\in Aut(G)$. In an earlier work we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group $G$ has the $R_\\infty$-property then $G^\\varphi$ is infinite for all $\\varphi\\in Aut(G)$. In this article we show that the condition is also sufficient. In particular we deduce that a Borel subgroup of any semisimple algebraic group has the $R_\\infty$-property and identify certain classes of solvable algebraic groups for which the property fails.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-08T10:39:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G07",
      "20E36"
    ],
    "keywords": [
      "linear algebraic group",
      "semisimple algebraic group",
      "algebraic group automorphisms",
      "property fails",
      "borel subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}