{
  "id": "2411.04800",
  "version": "v1",
  "published": "2024-11-07T15:38:32.000Z",
  "updated": "2024-11-07T15:38:32.000Z",
  "title": "Configuration spaces of circles in the plane",
  "authors": [
    "Justin Curry",
    "Ryan Gelnett",
    "Matthew C. B. Zaremsky"
  ],
  "comment": "27 pages, 13 figures",
  "categories": [
    "math.AT",
    "math.GR",
    "math.GT"
  ],
  "abstract": "We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these fundamental groups are obtained as iterated semdirect products of braid groups, with the structure for each component dictated by a finite rooted tree. These groups can be viewed as \"braided\" versions of the automorphism groups of such trees. We also discuss connections to statistical mechanics, topological data analysis, and geometric group theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-07T15:38:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55R80",
      "20F65",
      "20F36"
    ],
    "keywords": [
      "configuration spaces",
      "fundamental group",
      "geometric group theory",
      "braid groups",
      "finite rooted tree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}