Minimizing laminations in regular covers, horospherical orbit closures, and circle-valued Lipschitz maps

TL;DR

This paper explores the relationship between distance minimizing laminations and horospherical orbit closures in hyperbolic manifold covers, providing new constructions and analyzing the effects of metric perturbations on orbit closures.

Contribution

It introduces novel constructions of old covers with specific properties and explicitly describes all horocycle orbit closures, revealing sensitivity to metric changes.

Findings

Explicit descriptions of horocycle orbit closures in old covers.

Small metric perturbations cause significant topological changes.

New connections between laminations and orbit closures in hyperbolic geometry.

Abstract

We expose a connection between distance minimizing laminations and horospherical orbit closures in $\mathbb{Z}$-covers of compact hyperbolic manifolds. For surfaces, we provide novel constructions of $\mathbb{Z}$-covers with prescribed geometric and dynamical properties, in which an explicit description of all horocycle orbit closures is given. We further show that even the slightest of perturbations to the hyperbolic metric on a $\mathbb{Z}$-cover can lead to drastic topological changes to horocycle orbit closures.