{
  "id": "2505.07873",
  "version": "v1",
  "published": "2025-05-09T22:12:51.000Z",
  "updated": "2025-05-09T22:12:51.000Z",
  "title": "Solution to two open problems in geometric group theory",
  "authors": [
    "Jordan A. Sahattchieve"
  ],
  "comment": "Ph.D. dissertation, University of Michigan in Ann Arbor, 2012, 88 pages",
  "journal": "Groups Complex. Cryptol Vol. 7 No. 1 (2015); Serdica Math. J. 48 (2022)",
  "categories": [
    "math.GR"
  ],
  "abstract": "We introduce a method for analyzing the convex hull of a set in non-positively curved piecewise Euclidean polygonal complexes and we apply this method to prove that with the usual action of Fm x Zn on the metric product of a tree with Rn , every quasiconvex subgroup of Fm x Zn is convex. This answers the question whether a quasiconvex subgroup of a CAT(0) group is a CAT(0) group in the affirmative for the groups F m x Zn. We also prove bounded packing in a special class of polycyclic groups, and we introduce the notion of coset growth and provide a bound for the coset growth of uniform lattices in Sol.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-09T22:12:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F67",
      "20F16",
      "20F19"
    ],
    "keywords": [
      "geometric group theory",
      "open problems",
      "coset growth",
      "quasiconvex subgroup",
      "curved piecewise euclidean polygonal complexes"
    ],
    "tags": [
      "dissertation",
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 88,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}