Linear Slices of the Quasi-fuchsian Space of Punctured Tori

After fixing a marking (V,W ) of a quasi-Fuchsian punctured torus group G, the complex length λV and the complex twist τV,W parameters define a holomorphic embedding of the quasi-Fuchsian space QF of punctured tori into C2. It is called the complex Fenchel-Nielsen coordinates of QF . For c ∈ C, let Qγ,c be the affine subspace of C2 defined by the linear equation λV = c. Then we can consider the linear slice Lc of QF by QF ∩ Qγ,c which is a holomorphic slice of QF . For any positive real value c, Lc always contains the so-called Bers-Maskit slice BMγ,c defined in [Topology 43 (2004), no. 2, 447–491]. In this paper we show that if c is sufficiently small, then Lc coincides with BMγ,c whereas Lc has other components besides BMγ,c when c is sufficiently large. We also observe the scaling property of Lc.