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

    Research output: Contribution to journalArticle

    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 Jan 1

    Fingerprint

    Modular Group
    Riemann Surface
    Moduli Space
    Topology
    Quotient Space
    Geometric object
    Geometric Structure
    Complex Structure
    Set of points
    Completion
    Quotient

    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

    Cite this

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    title = "Dynamics of teichm{\"u}ller modular groups and topology of moduli spaces of Riemann surfaces of infinite type",
    abstract = "We investigate the dynamics of the Teichm{\"u}ller modular group on the Teichm{\"u}ller 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 Teichm{\"u}ller space. However, we introduce the set of points where the action of the Teichm{\"u}ller modular group is stable, and we prove that this region of stability is generic in the Teichm{\"u}ller space. By taking the quotient and completion with respect to the Teichm{\"u}ller 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.",
    keywords = "Hyperbolic geometry, Length spectrum, Limit set, Moduli space, Quasiconformal deformation, Region of discontinuity, Riemann surface of infinite type, Teichm{\"u}ller modular group",
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