{
  "id": "1106.0688",
  "version": "v5",
  "published": "2011-06-03T15:40:49.000Z",
  "updated": "2013-11-07T17:39:19.000Z",
  "title": "Spectral rigidity of automorphic orbits in free groups",
  "authors": [
    "Stefano Francaviglia",
    "Mathieu Carette",
    "Ilya Kapovich",
    "Armando Martino"
  ],
  "comment": "Included a corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of Out(F_N). Note that, because of the arXiv rules, the corrigendum and the original article are amalgamated into a single pdf file, with the corrigendum appearing first, followed by the main body of the original article",
  "journal": "Algebraic and Geometric Topology, vol. 12 (2012), pp. 1457--1486",
  "doi": "10.2140/agt.2012.12.1457",
  "categories": [
    "math.GR",
    "math.GT"
  ],
  "abstract": "It is well-known that a point $T\\in cv_N$ in the (unprojectivized) Culler-Vogtmann Outer space $cv_N$ is uniquely determined by its \\emph{translation length function} $||.||_T:F_N\\to\\mathbb R$. A subset $S$ of a free group $F_N$ is called \\emph{spectrally rigid} if, whenever $T,T'\\in cv_N$ are such that $||g||_T=||g||_{T'}$ for every $g\\in S$ then $T=T'$ in $cv_N$. By contrast to the similar questions for the Teichm\\\"uller space, it is known that for $N\\ge 2$ there does not exist a finite spectrally rigid subset of $F_N$. In this paper we prove that for $N\\ge 3$ if $H\\le Aut(F_N)$ is a subgroup that projects to an infinite normal subgroup in $Out(F_N)$ then the $H$-orbit of an arbitrary nontrivial element $g\\in F_N$ is spectrally rigid. We also establish a similar statement for $F_2=F(a,b)$, provided that $g\\in F_2$ is not conjugate to a power of $[a,b]$. We also include an appended corrigendum which gives a corrected proof of Lemma 5.1 about the existence of a fully irreducible element in an infinite normal subgroup of of $Out(F_N)$. Our original proof of Lemma 5.1 relied on a subgroup classification result of Handel-Mosher, originally stated by Handel-Mosher for arbitrary subgroups $H\\le Out(F_N)$. After our paper was published, it turned out that the proof of the Handel-Mosher subgroup classification theorem needs the assumption that $H$ be finitely generated. The corrigendum provides an alternative proof of Lemma~5.1 which uses the corrected, finitely generated, version of the Handel-Mosher theorem and relies on the 0-acylindricity of the action of $Out(F_N)$ on the free factor complex (due to Bestvina-Mann-Reynolds). A proof of 0-acylindricity is included in the corrigendum.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2013-11-07T17:39:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free group",
      "spectral rigidity",
      "automorphic orbits",
      "infinite normal subgroup",
      "handel-mosher subgroup classification theorem needs"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.0688F"
    }
  }
}