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Inventiones mathematicae

, Volume 218, Issue 1, pp 197–299 | Cite as

\(\mathrm {SO}(p,q)\)-Higgs bundles and Higher Teichmüller components

  • Marta Aparicio-Arroyo
  • Steven Bradlow
  • Brian Collier
  • Oscar García-PradaEmail author
  • Peter B. Gothen
  • André Oliveira
Article
  • 97 Downloads

Abstract

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such ‘exotic’ components in moduli spaces of \(\mathrm {SO}(p,q)\)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group \(\mathrm {SO}(p,q)\). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for \(\mathrm {SO}(2,q)\), with \(q\geqslant 4\)).

Mathematics Subject Classification

14D20 14F45 14H60 

Notes

Acknowledgements

The authors are grateful to Olivier Guichard, Beatrice Pozzetti, Carlos Simpson, Richard Wentworth and Anna Wienhard for useful conversations and to the referee for a careful reading and for a number of helpful remarks and corrections.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Marta Aparicio-Arroyo
    • 1
  • Steven Bradlow
    • 2
  • Brian Collier
    • 3
  • Oscar García-Prada
    • 4
    Email author
  • Peter B. Gothen
    • 5
  • André Oliveira
    • 5
    • 6
  1. 1.Raet—HR software and servicesAlcobendasSpain
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Department of MathematicsUniversity of MarylandCollege ParkUSA
  4. 4.Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCMMadridSpain
  5. 5.Faculdade de Ciências da Universidade do PortoCentro de Matemática da Universidade do PortoPortoPortugal
  6. 6.Departamento de MatemáticaUniversidade de Trás-os-Montes e Alto Douro, UTADVila RealPortugal

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