Graph colorings and realization of manifolds as leaves

Abstract

This thesis has two main parts. The first one is devoted to show that, for any infinite connected (repetitive) graph X with finite maximum vertex degree D, there exists a (repetitive) limit-aperiodic coloring by at most D colors. Several direct consequences of this theorem are also derived, like the existence of (repetitive) limit-aperiodic colorings of any (repetitive) tiling of a Riemannian manifold. Another application is the proof of the existence of edge colorings with analogous properties. The second part is devoted to prove that any (repetitive) Riemannian manifold of bounded geometry can be isometrically realized as leaf of a compact Riemannian (minimal) foliated space, whose leaves have no holonomy. This also uses the previous result about colorings, but it also requires much more technical work concerning the space of pointed Riemannian manifolds with the topology defined by the smooth convergence.