Exotic components in linear slices of quasi-Fuchsian groups

TL;DR

This paper investigates the structure of linear slices of quasi-Fuchsian groups, proving the existence of non-standard components using new methods involving length functions and complex projective structures.

Contribution

It provides two novel proofs for the existence of non-standard components in linear slices, linking them to exotic projective structures with quasi-Fuchsian holonomy.

Findings

Existence of non-standard components for large length values

Characterization of non-standard components via exotic projective structures

New proofs based on length functions and complex earthquakes

Abstract

The linear slice of quasi-Fuchsian once-punctured torus groups is defined by fixing the complex length of some simple closed curve to be a fixed positive real number. It is known that the linear slice is a union of disks, and it always has one standard component containing Fuchsian groups. Komori and Yamashita proved that there exist non-standard components if the length is sufficiently large. We give two other proofs of their theorem, one is based on some properties of length functions, and the other is based on the theory of complex projective structures and complex earthquakes. From the latter proof, we can characterize the existence of non-standard components in terms of exotic projective structures with quasi-Fuchsian holonomy.