The nielsen realization problem for asymptotic teichm̈uller modular groups

Ege Fujikawa, Katsuhiko Matsuzaki

    Research output: Contribution to journalArticle

    Abstract

    Under a certain geometric assumption on a hyperbolic Riemann surface, we prove an asymptotic version of the fixed point theorem for the Teichm̈uller modular group, which asserts that every finite subgroup of the asymptotic Teichm̈uller modular group has a common fixed point in the asymptotic Teichm̈uller space. For its proof, we use a topological characterization of the asymptotically trivial mapping class group, which has been obtained in the authors' previous paper, but a simpler argument is given here. As a consequence, every finite subgroup of the asymptotic Teichm̈uller modular group is realized as a group of quasiconformal automorphisms modulo coincidence near infinity. Furthermore, every finite subgroup of a certain geometric automorphism group of the asymptotic Teichm̈uller space is realized as an automorphism group of the Royden boundary of the Riemann surface. These results can be regarded as asymptotic versions of the Nielsen realization theorem.

    Original languageEnglish
    Pages (from-to)3309-3327
    Number of pages19
    JournalTransactions of the American Mathematical Society
    Volume365
    Issue number6
    DOIs
    Publication statusPublished - 2013

    Fingerprint

    Modular Group
    Subgroup
    Riemann Surface
    Automorphism Group
    Hyperbolic Surface
    Quasiconformal
    Mapping Class Group
    Coincidence
    Common Fixed Point
    Fixed point theorem
    Modulo
    Automorphisms
    Trivial
    Infinity
    Theorem

    Keywords

    • Hyperbolic geometry
    • Quasiconformal mapping class group
    • Riemann surface
    • Teichmüller space

    ASJC Scopus subject areas

    • Mathematics(all)
    • Applied Mathematics

    Cite this

    The nielsen realization problem for asymptotic teichm̈uller modular groups. / Fujikawa, Ege; Matsuzaki, Katsuhiko.

    In: Transactions of the American Mathematical Society, Vol. 365, No. 6, 2013, p. 3309-3327.

    Research output: Contribution to journalArticle

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