{
  "id": "2407.04301",
  "version": "v1",
  "published": "2024-07-05T07:09:21.000Z",
  "updated": "2024-07-05T07:09:21.000Z",
  "title": "Limits of limit sets in rank-one symmetric spaces",
  "authors": [
    "Antonin Guilloux",
    "Theodore Weisman"
  ],
  "categories": [
    "math.GT"
  ],
  "abstract": "We consider the question of continuity of limit sets for sequences of geometrically finite subgroups of isometry groups of rank-one symmetric spaces, and prove analogues of classical (Kleinian) theorems in this context. In particular we show that, assuming strong convergence of the sequence of subgroups, the limit sets vary continuously with respect to Hausdorff distance, and if the sequence is weakly type-preserving, the sequence of Cannon-Thurston maps also converges uniformly to a limiting Cannon-Thurston map. Our approach uses the theory of extended geometrically finite representations, developed recently by the second author.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-05T07:09:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M50"
    ],
    "keywords": [
      "rank-one symmetric spaces",
      "limit sets vary",
      "finite subgroups",
      "isometry groups",
      "extended geometrically finite representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}