Compact deformations of Fuchsian groups

Christopher J. Bishop; Peter W. Jones

doi:10.1007/BF02868468

Compact deformations of Fuchsian groups

Christopher J. Bishop
, Peter W. Jones

Mathematics

Stony Brook University

Yale University

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

A conformal map Φ on the unit disk is called a compact deformation of a Fuchsian group G if Φ has a quasiconformal extension to the plane h which conjugates G to a Kleinian group G′ and the dilatation of h is compactly supported modulo G. We show that for such deformations δ = dim(Λ(G′)) = dim(Λ_c(G′)) (if δ ≥ 1) and the image of Λ_e = Λ \ Λ_c is contained in a countable union of rectifiable curves and has zero length iff G is divergence type.

Original language	English
Article number	BF02868468
Pages (from-to)	5-36
Number of pages	32
Journal	Journal d'Analyse Mathématique
Volume	87
Issue number	1
DOIs	https://doi.org/10.1007/BF02868468
State	Published - 2002

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title = "Compact deformations of Fuchsian groups",

abstract = "A conformal map Φ on the unit disk is called a compact deformation of a Fuchsian group G if Φ has a quasiconformal extension to the plane h which conjugates G to a Kleinian group G′ and the dilatation of h is compactly supported modulo G. We show that for such deformations δ = dim(Λ(G′)) = dim(Λc(G′)) (if δ ≥ 1) and the image of Λe = Λ \textbackslash{} Λc is contained in a countable union of rectifiable curves and has zero length iff G is divergence type.",

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AB - A conformal map Φ on the unit disk is called a compact deformation of a Fuchsian group G if Φ has a quasiconformal extension to the plane h which conjugates G to a Kleinian group G′ and the dilatation of h is compactly supported modulo G. We show that for such deformations δ = dim(Λ(G′)) = dim(Λc(G′)) (if δ ≥ 1) and the image of Λe = Λ \ Λc is contained in a countable union of rectifiable curves and has zero length iff G is divergence type.

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DO - 10.1007/BF02868468

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Compact deformations of Fuchsian groups

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