Small curvature laminations in hyperbolic 3-manifolds

TL;DR

This paper investigates the properties of small curvature laminations in hyperbolic 3-manifolds, establishing conditions under which boundary leaves have nontrivial or noncyclic fundamental groups, contributing to the understanding of lamination topology.

Contribution

It proves that sufficiently small curvature laminations with certain conditions have boundary leaves with nontrivial or noncyclic fundamental groups, providing new insights into lamination structure in hyperbolic 3-manifolds.

Findings

Boundary leaves have nontrivial fundamental groups under small curvature conditions.

Existence of a universal curvature bound ensuring boundary leaves have noncyclic fundamental groups.

Abstract

We show that if $\mathcal{L}$ is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of $\mathcal{L}$ are all in the interval $(-\delta ,\delta)$ for a fixed $\delta\in[0,1)$ and no complimentary region of $\mathcal{L}$ is an interval bundle over a surface, then each boundary leaf of $\mathcal{L}$ has a nontrivial fundamental group. We also prove existence of a fixed constant $\delta_0 > 0$ such that if $\mathcal{L}$ is a codimension-one lamination in a finite volume hyperbolic 3-manifold such that the principal curvatures of each leaf of $\mathcal{L}$ are all in the interval $(-\delta_0 ,\delta_0)$ and no complimentary region of $\mathcal{L}$ is an interval bundle over a surface, then each boundary leaf of $\mathcal{L}$ has a noncyclic fundamental group.