Quasiconformal mappings of \(Y\)-pieces. (English) Zbl 1064.30045

Bishop, Christopher J.

Rev. Mat. Iberoam. 18, No. 3, 627-652 (2002).

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.

Reviewer: Gou Nakamura (Toyota)

MSC:

30F60	Teichmüller theory for Riemann surfaces
30C20	Conformal mappings of special domains

Keywords:

Y-pieces; Hausdorff dimension; quasi-Fuchsian groups; quasiconformal deformation, critical exponent; convex core

Citations:

Zbl 1064.30041

References:

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[2]	Beardon, A.F.: The geometry of discrete groups. Springer-Verlag, New York, 1983. · Zbl 0528.30001
[3]	Bishop, C.J.: Non-rectifiable limit sets of dimension one. Rev. Mat. Iberoamericana 18 (2002), 653-684. · Zbl 1064.30046
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[5]	Bishop, C.J.: Divergence groups have the Bowen property. Ann. of Math. 154 (2001), 205-217. · Zbl 0999.37030
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[7]	Bishop, C.J.: \delta -stable Fuchsian groups. Preprint, 2002. · Zbl 1027.30064