Reduced Bers boundaries of Teichmüller spaces

Ken’ichi Ohshika[1]

  • [1] Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 1, page 145-176
  • ISSN: 0373-0956

Abstract

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We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space lying there into a point.This boundary turns out to be independent of the basepoint, and the action of the mapping class group extends continuously to this boundary.This is an affirmative answer to Thurston’s conjecture.He also conjectured that this boundary is homeomorphic to the unmeasured lamination space by the correspondence coming from ending laminations.This part of the conjecture needs some correction: we show that the quotient topology of the reduced Bers boundary is different form the topology induced from the unmeasured lamination space.Furthermore, we show that every auto-homeomorphism on the reduced Bers boundary comes from a unique extended mapping class.We also give a way to determine the limit in the reduced Bers boundary for a given sequence in Teichmüller space.

How to cite

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Ohshika, Ken’ichi. "Reduced Bers boundaries of Teichmüller spaces." Annales de l’institut Fourier 64.1 (2014): 145-176. <http://eudml.org/doc/275592>.

@article{Ohshika2014,
abstract = {We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space lying there into a point.This boundary turns out to be independent of the basepoint, and the action of the mapping class group extends continuously to this boundary.This is an affirmative answer to Thurston’s conjecture.He also conjectured that this boundary is homeomorphic to the unmeasured lamination space by the correspondence coming from ending laminations.This part of the conjecture needs some correction: we show that the quotient topology of the reduced Bers boundary is different form the topology induced from the unmeasured lamination space.Furthermore, we show that every auto-homeomorphism on the reduced Bers boundary comes from a unique extended mapping class.We also give a way to determine the limit in the reduced Bers boundary for a given sequence in Teichmüller space.},
affiliation = {Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan},
author = {Ohshika, Ken’ichi},
journal = {Annales de l’institut Fourier},
keywords = {Bers boundary; Teichmüller space; Kleinian group; Teichüller space},
language = {eng},
number = {1},
pages = {145-176},
publisher = {Association des Annales de l’institut Fourier},
title = {Reduced Bers boundaries of Teichmüller spaces},
url = {http://eudml.org/doc/275592},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Ohshika, Ken’ichi
TI - Reduced Bers boundaries of Teichmüller spaces
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 1
SP - 145
EP - 176
AB - We consider a quotient space of the Bers boundary of Teichmüller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space lying there into a point.This boundary turns out to be independent of the basepoint, and the action of the mapping class group extends continuously to this boundary.This is an affirmative answer to Thurston’s conjecture.He also conjectured that this boundary is homeomorphic to the unmeasured lamination space by the correspondence coming from ending laminations.This part of the conjecture needs some correction: we show that the quotient topology of the reduced Bers boundary is different form the topology induced from the unmeasured lamination space.Furthermore, we show that every auto-homeomorphism on the reduced Bers boundary comes from a unique extended mapping class.We also give a way to determine the limit in the reduced Bers boundary for a given sequence in Teichmüller space.
LA - eng
KW - Bers boundary; Teichmüller space; Kleinian group; Teichüller space
UR - http://eudml.org/doc/275592
ER -

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