Linear slices of the quasifuchsian space of punctured tori

Details Linear slices of the quasifuchsian space of punctured tori Linear slices of the quasifuchsian space of punctured tori

2011-11-15

arxiv.org/abs/1111.3407v1

Conform. Geom. Dyn. 16 (2012), 89-102

Mathematics

Geometric Topology

15 pages, 8 figures. arXiv admin note: some text overlap with arXiv:math/0209189

Scientific paper

10.1090/S1088-4173-2012-00237-8

After fixing a marking (V, W) of a quasifuchsian punctured torus group G, the complex length l_V and the complex twist tau_V,W parameters define a holomorphic embedding of the quasifuchsian space QF of punctured tori into C^2. It is called the complex Fenchel-Nielsen coordinates of QF. For a complex number c, let Q_gamma,c be the affine subspace of C^2 defined by the linear equation l_V=c. Then we can consider the linear slice L of QF by QF \cap Q_gamma,c which is a holomorphic slice of QF. For any positive real value c, L always contains the so called Bers-Maskit slice BM_gamma,c. In this paper we show that if c is sufficiently small, then L coincides with BM_gamma,c whereas L has other components besides BM_gamma,c when c is sufficiently large. We also observe the scaling property of L.

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