Quasiconformal mappings of \(Y\)-pieces.
type: Item
instance of: scholarly article
title: Quasiconformal mappings of \(Y\)-pieces.
zbMATH Open document ID: 1064.30045
DOI: 10.4171/RMI/330
author: Christopher J. Bishop
published in: Revista Matemática Iberoamericana
publication date: 2003-09-25T00:00:00Z
full work available at URL: https://eudml.org/doc/39697
review text: In order to deform a Riemann surface by shortening a closed geodesic, the author gives a quasiconformal mapping between \(Y\)-pieces. A Riemann surface with boundary is called a generalized \(Y\)-piece if it is homeomorphic to a 2-sphere minus three disks (or points) and the boundary consists of three closed geodesics (or punctures). When \(Y\)-pieces \(Y_1\), \(Y_2\) have three boundary lengths \((a_1,b_1,c_1)\), \((a_2,b_1,c_1)\), respectively, a quasiconformal mapping \(f:Y_1\to Y_2\) is considered such that it is isometric on two boundary components \(\gamma_b\), \(\gamma_c\) and multiplies length by \(a_2/a_1\) on \(\gamma_a\). Furthermore the Beltrami coefficient of \(f\) decays exponentially according to the distance from \(\gamma_a\). The quasiconformal mappings in this paper are used in another paper by the author [Rev. Mat. Iberoam. 18, No. 3, 653--684 (2002; Zbl 1064.30041)] to construct quasi-Fuchsian groups whose limit sets are non-rectifiable curves of dimension 1.
reviewed by: Gou Nakamura
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 30C20
zbMATH DE Number: 1931493
zbMATH Keywords: Y-pieces
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: quasi-Fuchsian groups
zbMATH Keywords: quasiconformal deformation, critical exponent
zbMATH Keywords: convex core
MaRDI profile type: Publication
cites work: Q3310101
cites work: Non-rectifiable limit sets of dimension one.
cites work: Divergence groups have the Bowen property
cites work: Big deformations near infinity
cites work: Q4429908
OpenAlex ID: W1992351120
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description: scientific article; zbMATH DE number 1931493