Dynamics of teichmüller modular groups and topology of moduli spaces of Riemann surfaces of infinite type

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Abstract

We investigate the dynamics of the Teichmüller modular group on the Teichmüller space of a Riemann surface of infinite topological type. Since the modular group does not necessarily act discontinuously, the quotient space cannot inherit a rich geometric structure from the Teichmüller space. However, we introduce the set of points where the action of the Teichmüller modular group is stable, and we prove that this region of stability is generic in the Teichmüller space. By taking the quotient and completion with respect to the Teichmüller distance, we obtain a geometric object that we regard as an appropriate moduli space of the quasiconformally equivalent complex structures admitted on a topologically infinite Riemann surface.

Original languageEnglish
Pages (from-to)1-64
Number of pages64
JournalGroups, Geometry, and Dynamics
Volume12
Issue number1
DOIs
Publication statusPublished - 2018

Keywords

  • Hyperbolic geometry
  • Length spectrum
  • Limit set
  • Moduli space
  • Quasiconformal deformation
  • Region of discontinuity
  • Riemann surface of infinite type
  • Teichmüller modular group

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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