Linear slices of the quasi-Fuchsian space of punctured tori

Abstract:

After fixing a marking $(V, W)$ of a quasi-Fuchsian punctured torus group $G$, the complex length $\lambda _V$ and the complex twist $\tau _{V,W}$ parameters define a holomorphic embedding of the quasi-Fuchsian space ${\mathcal Q}{\mathcal F}$ of punctured tori into $\mathbf {C}^2$. It is called the complex Fenchel-Nielsen coordinates of $\mathcal {QF}$. For $c \in \mathbf {C}$, let $\mathcal {Q}_{\gamma , c}$ be the affine subspace of $\mathbf {C}^2$ defined by the linear equation $\lambda _V=c$. Then we can consider the linear slice ${\mathcal L}_c$ of $\mathcal {QF}$ by $\mathcal {QF} \cap \mathcal {Q}_{\gamma , c}$ which is a holomorphic slice of $\mathcal {QF}$. For any positive real value $c$, ${\mathcal L}_c$ always contains the so-called Bers-Maskit slice ${\mathcal BM}_{\gamma , c}$ defined in [Topology 43 (2004), no. 2, 447–491]. In this paper we show that if $c$ is sufficiently small, then ${\mathcal L}_c$ coincides with ${\mathcal BM}_{\gamma , c}$ whereas ${\mathcal L}_c$ has other components besides ${\mathcal BM}_{\gamma , c}$ when $c$ is sufficiently large. We also observe the scaling property of ${\mathcal L}_c$.