JournalsrmiVol. 36, No. 7pp. 2121–2146

Discrete dynamics and differentiable stacks

  • Alejandro Cabrera

    Universidade Federal do Rio de Janeiro, Brazil
  • Matias L. del Hoyo

    Universidade Federal Fluminense, Niterói, Brazil
  • Enrique Pujals

    City University of New York, USA
Discrete dynamics and differentiable stacks cover
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In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffel’s theorem on irrational rotations of the circle, we compute the stack-theoretic fundamental group of hyperbolic toral automorphisms, and we revisit the classification of lens spaces.

Cite this article

Alejandro Cabrera, Matias L. del Hoyo, Enrique Pujals, Discrete dynamics and differentiable stacks. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2121–2146

DOI 10.4171/RMI/1194