{
  "id": "2204.03278",
  "version": "v1",
  "published": "2022-04-07T08:04:03.000Z",
  "updated": "2022-04-07T08:04:03.000Z",
  "title": "Finiteness properties of some groups of piecewise projective homeomorphisms",
  "authors": [
    "Daniel Farley"
  ],
  "comment": "45 pages; 6 Figures; 4 Tables",
  "categories": [
    "math.GR"
  ],
  "abstract": "The Lodha-Moore group $G$ first arose as a finitely presented counterexample to von Neumann's conjecture. The group $G$ acts on the unit interval via piecewise projective homemorphisms. A result of Lodha shows that $G$ in fact has type $F_{\\infty}$. Here we will describe $G$ as a group that is \"locally determined\" by an inverse semigroup $S_{2}$, in the sense of the author's joint work with Hughes. The semigroup $S_{2}$ is generated by three linear fractional transformations $A$, $B$, and $C_{2}$, where $A$ and $B$ are elliptical transformations of the hyperbolic plane and $C_{2}$ is a hyperbolic translation. Following a general procedure delineated by Farley and Hughes, we offer a new proof that $G$ has type $F_{\\infty}$. Our proof simultaneously shows that various groups acting on the line, the circle, and the Cantor set have type $F_{\\infty}$. We also prove analogous results for the groups that are locally determined by an inverse semigroup $S_{3}$, which shares the generators $A$ and $B$ with $S_{2}$, but replaces $C_{2}$ with a different hyperbolic translation $C_{3}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-07T08:04:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20J05",
      "20M18"
    ],
    "keywords": [
      "piecewise projective homeomorphisms",
      "finiteness properties",
      "hyperbolic translation",
      "inverse semigroup",
      "linear fractional transformations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}