Measured foliations at infinity of quasi-Fuchsian manifolds

TL;DR

This paper investigates the measured foliations at infinity of quasi-Fuchsian manifolds, showing they are filling near the Fuchsian locus and can be realized as such for scaled pairs, answering existing questions in the field.

Contribution

It establishes that measured foliations at infinity are filling near the Fuchsian locus and demonstrates the realization of scaled filling pairs as boundary data of quasi-Fuchsian manifolds.

Findings

Measured foliations at infinity are filling if the manifold is close to Fuchsian.

Any scaled filling pair of measured foliations can be realized at infinity of some quasi-Fuchsian manifold.

Answers to Schlenker's questions near the Fuchsian locus are provided.

Abstract

Let $(\lambda^+(M),\lambda^-(M))$ denote the pair of measured foliations at the boundary at infinity $\partial_\infty$ of a quasi-Fuchsian manifold $M$. We prove that $(\lambda^+(M),\lambda^-(M))$ is filling if $M$ is close to being Fuchsian. We also show that given any filling pair $(\alpha_1,\alpha_2)$ of measured foliations, and every small enough $t>0$, the pair $(t\alpha_1,t\alpha_2)$ is realised as the pair of measured foliations at infinity of some quasi-Fuchsian manifold $M$. This answers questions of Schlenker near the Fuchsian locus.