Linear slices close to a Maskit slice

TL;DR

This paper investigates the behavior of linear slices of Kleinian once-punctured torus groups near the Maskit slice, revealing different convergence properties depending on the manner of trace convergence and discovering a non-locally connected slice.

Contribution

It characterizes the convergence of linear slices to the Maskit slice under different trace convergence modes and shows the existence of a non-locally connected linear slice.

Findings

Linear slices converge to the Maskit slice when traces converge horocyclically to 2.

Linear slices converge to a subset of the Maskit slice when traces converge tangentially to 2.

A linear slice that is not locally connected is constructed.

Abstract

We consider linear slices of the space of Kleinian once-punctured torus groups; a linear slice is obtained by fixing the value of the trace of one of the generators. The linear slice for trace 2 is called the Maskit slice. We will show that if traces converge `horocyclically' to 2 then associated linear slices converge to the Maskit slice, whereas if the traces converge `tangentially' to 2 the linear slices converge to a proper subset of the Maskit slice. This result will be also rephrased in terms of complex Fenchel-Nielsen coordinates. In addition, we will show that there is a linear slice which is not locally connected.