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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 4, pp 1247–1268 | Cite as

Teichmüller theory and collapse of flat manifolds

  • Renato G. BettiolEmail author
  • Andrzej Derdzinski
  • Paolo Piccione
Article
First Online:

Abstract

We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by collapsing closed flat manifolds, and the collapsed limits of closed flat 3-manifolds are classified.

Keywords

Flat manifolds Flat orbifolds Teichmueller space Moduli space Collapse Gromov–Hausdorff convergence 

Mathematics Subject Classification

22E40 32G15 53C15 53C24 53C29 57M60 57R18 58D17 58D27 
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Notes

Acknowledgements

It is a pleasure to thank Alexander Lytchak, John Harvey, Karsten Grove, and Curtis Pro for suggestions regarding equivariant Gromov–Hausdorff convergence, Burkhard Wilking for comments on realizing flat orbifolds as limits of flat manifolds, and Andrzej Szczepański for conversations about the holonomy representation of flat manifolds. Part of this work was done during a visit of the first named author to the Max Planck Institute for Mathematics in Bonn, Germany, and of the third named author to the University of Notre Dame, USA; they would like to thank these institutions for providing excellent working conditions. The second and third named authors are partially supported by a FAPESP-OSU 2015 Regular Research Award (FAPESP Grant: 2015/50315-3).

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Renato G. Bettiol
    • 1
    Email author
  • Andrzej Derdzinski
    • 2
  • Paolo Piccione
    • 3
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA
  3. 3.Departamento de MatemáticaUniversidade de São PauloSão PauloBrazil

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