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 Q_{γ, c} be the affine subspace of C² defined by the linear equation λV = c. Then we can consider the linear slice L_c of QF by QF ∩ Q_{γ, c} which is a holomorphic slice of QF. For any positive real value c, L_c always contains the so-called Bers-Maskit slice BM_{γ, c} defined in [Topology 43 (2004), no. 2, 447–491]. In this paper we show that if c is sufficiently small, then L_c coincides with BM_{γ, c} whereas L_c has other components besides BM_{γ, c} when c is sufficiently large. We also observe the scaling property of L_c.