Annales Academić Scientiarum Fennicć
Mathematica
Volumen 35, 2010, 275-283

AMBIENT QUASICONFORMAL HOMOGENEITY OF PLANAR DOMAINS

Petra Bonfert-Taylor, Richard D. Canary, Gaven Martin, Edward C. Taylor and Michael Wolf

Wesleyan University, Department of Mathematics
265 Church Street, Middletown, CT 06459, U.S.A.; pbonfert 'at' wesleyan.edu

University of Michigan, Department of Mathematics
2074 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, U.S.A.; canary 'at' umich.edu

Massey University, Institute of Information and Mathematical Sciences
Albany, Auckland, New Zealand; g.j.martin 'at' massey.ac.nz

Wesleyan University, Department of Mathematics
265 Church Street, Middletown, CT 06459, U.S.A.; ectaylor 'at' wesleyan.edu

Rice University, Department of Mathematics
P.O. Box 1892, Houston, TX 77251, U.S.A.; mwolf 'at' math.rice.edu

Abstract. We prove that the ambient quasiconformal homogeneity constant of a hyperbolic planar domain which is not simply connected is uniformly bounded away from 1.

We also consider a component \Omega₀ of the domain of discontinuity of a finitely generated Kleinian group \Gamma. We show that if \Omega₀/\Gamma is compact, then \Omega₀ is uniformly ambiently quasiconformally homogeneous, and that if \Omega₀ is not simply connected and its quotient \Omega₀/\Gamma is non-compact, then \Omega₀ is not uniformly quasiconformally homogeneous.

2000 Mathematics Subject Classification: Primary 30C62; Secondary 30F45.

Key words: Quasiconformal homogeneity, hyperbolic planar domain, Kleinian groups, quasiconformal removability.

Reference to this article: P. Bonfert-Taylor, R.D. Canary, G. Martin, E.C. Taylor and M. Wolf: Ambient quasiconformal homogeneity of planar domains. Ann. Acad. Sci. Fenn. Math. 35 (2010), 275-283.

Full document as PDF file

doi:10.5186/aasfm.2010.3516