Growth and homogeneity of matchbox manifolds

TL;DR

This paper investigates matchbox manifolds with equicontinuous holonomy, showing that polynomial growth of leaves constrains their structure, leading to homogeneity in certain cases and revealing obstructions in others.

Contribution

It establishes conditions under which matchbox manifolds with equicontinuous holonomy are homogeneous, especially relating leaf growth rates to the structure of their holonomy groups.

Findings

Matchbox manifolds with polynomial leaf growth of degree ≤ 3 are finite covers of homogeneous spaces.

For leaf growth of degree ≥ 4, obstructions to homogeneity are linked to nilpotent group structures.

Abstract

A matchbox manifold with one-dimensional leaves which has equicontinuous holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a classical Vietoris solenoid. In this work, we consider the problem, what can be said about a matchbox manifold with equicontinuous holonomy dynamics, and all of whose leaves have at most polynomial growth type? We show that such a space must have a finite covering for which the global holonomy group of its foliation is nilpotent. As a consequence, we show that if the growth type of the leaves is polynomial of degree at most 3, then there exists a finite covering which is homogeneous. If the growth type of the leaves is polynomial of degree at least 4, then there are additional obstructions to homogeneity, which arise from the structure of nilpotent groups.