{
  "id": "2305.00610",
  "version": "v1",
  "published": "2023-05-01T00:30:34.000Z",
  "updated": "2023-05-01T00:30:34.000Z",
  "title": "Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank",
  "authors": [
    "Mikolaj Fraczyk",
    "Minju Lee"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DS",
    "math.GT"
  ],
  "abstract": "Let $G$ be a connected semisimple real algebraic group and $\\Gamma<G$ be its Zariski dense discrete subgroup. We prove that if $\\Gamma\\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $\\Gamma$ is virtually a product of higher rank lattices and discrete subgroups of rank one factors of $G$. This may be viewed as a measure-theoretic analogue of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. The key ingredients in our proof are the product structure of leafwise measures and the high entropy method of Einsiedler-Katok-Lindenstrauss. In a companion paper jointly with Edwards and Oh, we use this result to show that the bottom of the $L^2$ spectrum has no atom in any infinite volume quotient of a higher rank simple algebraic group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-01T00:30:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A17"
    ],
    "keywords": [
      "finite bowen-margulis-sullivan measure",
      "discrete subgroup",
      "higher rank simple algebraic group",
      "connected semisimple real algebraic group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}