Discrete dynamics and differentiable stacks

Alejandro Cabrera ^[1] ; Matias L. del Hoyo ^[2] ; Enrique Pujals ^[3]
1. [1] Universidade Federal do Rio de Janeiro
  
  Universidade Federal do Rio de Janeiro
  
  Brasil
2. [2] Universidade Federal Fluminense
  
  Universidade Federal Fluminense
  
  Brasil
3. [3] City University of New York
  
  City University of New York
  
  Estados Unidos
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Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 7, 2020, págs. 2121-2146
Idioma: inglés
DOI: 10.4171/rmi/1194
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Resumen
- 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.