Realization of manifolds as leaves using graph colorings

TL;DR

This paper demonstrates that any repetitive Riemannian manifold with bounded geometry can be embedded as a leaf in a minimal Riemannian matchbox manifold without holonomy, using graph coloring techniques.

Contribution

It introduces a method to realize Riemannian manifolds as leaves in matchbox manifolds, extending previous results to include bounded geometry and minimality.

Findings

Any repetitive Riemannian manifold of bounded geometry can be realized as a leaf.

The methods can be adapted for Cantor transversals and prescribed holonomy coverings.

Realization as a dense leaf may not be possible with certain adaptations.

Abstract

It is proved that any (repetitive) Riemannian manifold of bounded geometry can be realized as a leaf of some (minimal) Riemannian matchbox manifold without holonomy. Our methods can be adapted to achieve Cantor transversals or a prescribed holonomy covering, but then the manifold may not be realized as a dense leaf.