Geometry & Topology

Extending homeomorphisms of the circle to quasiconformal homeomorphisms of the disk

David Epstein and Vladimir Markovic

Full-text: Open access


We prove that it is not possible to extend, in a homomorphic fashion, each quasisymmetric homeomorphism of the circle to a quasiconformal homeomorphism of the disk.

Article information

Geom. Topol., Volume 11, Number 1 (2007), 517-595.

Received: 28 September 2006
Accepted: 14 February 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane 37E10: Maps of the circle 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 57M50: Geometric structures on low-dimensional manifolds 57M60: Group actions in low dimensions

quasiconformal quasisymmetric circle low-dimensional manifolds group actions in low dimensions quasiconformal mappings in the plane maps of the circle homeomorphisms and diffeomorphisms of planes and surfaces


Epstein, David; Markovic, Vladimir. Extending homeomorphisms of the circle to quasiconformal homeomorphisms of the disk. Geom. Topol. 11 (2007), no. 1, 517--595. doi:10.2140/gt.2007.11.517.

Export citation


  • L V Ahlfors, Lectures on quasiconformal mappings, Manuscript prepared with the assistance of C J Earle, Jr., Van Nostrand Mathematical Studies 10 (1966)
  • A Beurling, L Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956) 125–142
  • A Casson, D Jungreis, Convergence groups and Seifert fibered $3$-manifolds, Invent. Math. 118 (1994) 441–456
  • A Douady, C J Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986) 23–48
  • D B A Epstein, Functors between tensored categories, Invent. Math. 1 (1966) 221–228
  • D B A Epstein, A Marden, V Markovic, Quasiconformal homeomorphisms and the convex hull boundary, Ann. of Math. $(2)$ 159 (2004) 305–336
  • D B A Epstein, W P Thurston, Transformation groups and natural bundles, Proc. London Math. Soc. (3) 38 (1979) 219–236
  • D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. $(2)$ 136 (1992) 447–510
  • F P Gardiner, D P Sullivan, Symmetric structures on a closed curve, Amer. J. Math. 114 (1992) 683–736
  • É Ghys, Prolongements des difféomorphismes de la sphère, Enseign. Math. $(2)$ 37 (1991) 45–59
  • J A Giblin, On the structure of homeomorphism groups, PhD thesis, University of Warwick (2007)
  • A Hinkkanen, Abelian and nondiscrete convergence groups on the circle, Trans. Amer. Math. Soc. 318 (1990) 87–121
  • A Hinkkanen, The structure of certain quasisymmetric groups, Mem. Amer. Math. Soc. 83 (1990)
  • S P Kerckhoff, The Nielsen realization problem, Ann. of Math. $(2)$ 117 (1983) 235–265
  • B Knaster, C Kuratowski, Sur les ensembles connexes, Fund. Math. 2 (1922) 206–255
  • V Markovic, Realization of the mapping class group by homeomorphisms To be published in Invent. Math. (2007) Available at \setbox0\makeatletter\@url {\unhbox0
  • V Markovic, Quasisymmetric groups, J. Amer. Math. Soc. 19 (2006) 673–715
  • J N Mather, Integrability in codimension $1$, Comment. Math. Helv. 48 (1973) 195–233
  • S Morita, Geometry of characteristic classes, Translations of Mathematical Monographs 199, Amer. Math. Soc. (2001) Translated from the 1999 Japanese original, Iwanami Series in Modern Mathematics
  • J F Plante, Subgroups of continuous groups acting differentiably on the half-line, Ann. Inst. Fourier $($Grenoble$)$ 34 (1984) 47–56
  • D Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, from: “Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (Stony Brook, 1978)”, Ann. of Math. Stud. 97, Princeton Univ. Press (1981) 465–496
  • T Tsuboi, On the foliated products of class $C\sp 1$, Ann. of Math. $(2)$ 130 (1989) 227–271
  • P Tukia, On two-dimensional quasiconformal groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980) 73–78
  • P Tukia, Homeomorphic conjugates of Fuchsian groups, J. Reine Angew. Math. 391 (1988) 1–54