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

##### MSC:
 30F60 Teichmüller theory for Riemann surfaces 30C20 Conformal mappings of special domains
Zbl 1064.30041
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##### References:
 [1] Álvarez, V. and Rodríguez, J.M.: Structure theorems for topological and Riemann surfaces. To appear in J. London. Math. Soc. [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 [4] Bishop, C.J.: A criterion for the failure of Ruelle’s property. Preprint, 1999. [5] Bishop, C.J.: Divergence groups have the Bowen property. Ann. of Math. 154 (2001), 205-217. · Zbl 0999.37030 [6] Bishop, C.J.: Big deformations near infinity. Preprint, 2002. · Zbl 1040.30024 [7] Bishop, C.J.: \delta -stable Fuchsian groups. Preprint, 2002. · Zbl 1027.30064
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