{
  "id": "2404.02730",
  "version": "v1",
  "published": "2024-04-03T13:29:40.000Z",
  "updated": "2024-04-03T13:29:40.000Z",
  "title": "Embedding relatively hyperbolic groups into products of binary trees",
  "authors": [
    "Patrick S. Nairne"
  ],
  "comment": "Comments very welcome",
  "categories": [
    "math.GR"
  ],
  "abstract": "We prove that if a group $G$ is relatively hyperbolic with respect to virtually abelian peripheral subgroups then $G$ quasiisometrically embeds into a product of binary trees. This extends the result of Buyalo, Dranishnikov and Schroeder in which they prove that a hyperbolic group quasiisometrically embeds into a product of binary trees. To prove the main result, we use the machinery of projection complexes and quasi-trees of metric spaces developed by Bestvina, Bromberg, Fujiwara and Sisto. We build on this theory by proving that one can remove certain edges from the quasi-tree of metric spaces, and be left with a tree of metric spaces which is quasiisometric to the quasi-tree of metric spaces. In particular, this reproves a result of Hume. Further, inspired by Buyalo, Dranishnikov and Schroeder's Alice's Diary, we develop a general theory of diaries and linear statistics. These notions provide a framework by which one can take a quasiisometric embedding of a metric space into a product of infinite-valence trees and upgrade it to a quasiisometric embedding into a product of binary trees.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-03T13:29:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "51F30",
      "68R15",
      "30L05"
    ],
    "keywords": [
      "embedding relatively hyperbolic groups",
      "binary trees",
      "metric space",
      "virtually abelian peripheral subgroups",
      "hyperbolic group quasiisometrically embeds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}