Linear slices of the quasifuchsian space of punctured tori

TL;DR

This paper investigates the structure of linear slices of the quasifuchsian space of punctured tori, revealing conditions under which these slices coincide with Bers-Maskit slices and exploring their scaling properties.

Contribution

It characterizes the linear slices of quasifuchsian space in terms of complex Fenchel-Nielsen coordinates and identifies when these slices match Bers-Maskit slices.

Findings

For small c, the linear slice equals the Bers-Maskit slice.

For large c, the linear slice has additional components.

The linear slices exhibit a scaling property.

Abstract

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.