Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalJournal d'Analyse Mathematique
Volume102
Issue number1
DOIs
Publication statusPublished - 2007 Aug
Externally publishedYes

Fingerprint

Quasiconformal Mapping
Mapping Class Group
Common Fixed Point
Riemann Surface
Fiber
Countable
Subspace
Projection
Imply
Generalization

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis

Cite this

@article{0eb16e51381b420b8b35811e70ac04dc,
title = "Quasiconformal mapping class groups having common fixed points on the asymptotic Teichm{\"u}ller spaces",
abstract = "For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichm{\"u}ller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichm{\"u}ller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.",
author = "Katsuhiko Matsuzaki",
year = "2007",
month = "8",
doi = "10.1007/s11854-007-0015-z",
language = "English",
volume = "102",
pages = "1--28",
journal = "Journal d'Analyse Mathematique",
issn = "0021-7670",
publisher = "Springer New York",
number = "1",

}

TY - JOUR

T1 - Quasiconformal mapping class groups having common fixed points on the asymptotic Teichmüller spaces

AU - Matsuzaki, Katsuhiko

PY - 2007/8

Y1 - 2007/8

N2 - For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.

AB - For an analytically infinite Riemann surface R, we consider the action of the quasiconformal mapping class group MCG(R) on the Teichmüller space T(R), which preserves the fibers of the projection α: T(R) → AT(R) onto the asymptotic Teichmüller space AT(R). We prove that if MCG(R) has a common fixed point α(p) AT(R), then it acts discontinuously on the fiber T p over α(p), which is a separable subspace of T(R). In particular, this implies that MCG(R) is a countable group. This is a generalization of a fact that MCG(R) acts discontinuously on T o = T(R) for an analytically finite Riemann surface R.

UR - http://www.scopus.com/inward/record.url?scp=58449100711&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=58449100711&partnerID=8YFLogxK

U2 - 10.1007/s11854-007-0015-z

DO - 10.1007/s11854-007-0015-z

M3 - Article

VL - 102

SP - 1

EP - 28

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

IS - 1

ER -