{
  "id": "1910.04620",
  "version": "v1",
  "published": "2019-10-10T14:57:42.000Z",
  "updated": "2019-10-10T14:57:42.000Z",
  "title": "Small $C^1$ actions of semidirect products on compact manifolds",
  "authors": [
    "Christian Bonatti",
    "Sang-hyun Kim",
    "Thomas Koberda",
    "Michele Triestino"
  ],
  "comment": "10 pages",
  "categories": [
    "math.GT",
    "math.DS",
    "math.GR"
  ],
  "abstract": "Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\\psi^*$ on $H^1(S,\\mathbb{R})$ has no eigenvalues on the unit circle, then there exists a neighborhood $\\mathcal U$ of the trivial action in the space of $C^1$ actions of $\\pi_1(T)$ on $M$ such that any action in $\\mathcal{U}$ is abelian. We will prove that the same result holds in the generality of an infinite cyclic extension of an arbitrary finitely generated group $H$, provided that the conjugation action of the cyclic group on $H^1(H,\\mathbb{R})\\neq 0$ has no eigenvalues of modulus one. We thus generalize a result of A. McCarthy, which addressed the case of abelian--by--cyclic groups acting on compact manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-10T14:57:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "compact manifolds",
      "semidirect products",
      "compact riemannian manifold",
      "infinite cyclic extension",
      "eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}