Low-dimensional solenoidal manifolds

TL;DR

This paper surveys low-dimensional solenoidal manifolds, exploring their structure, properties, and relation to inverse limits of finite covers, with new results on their laminated and principal bundle structures.

Contribution

It provides a comprehensive survey of 1-3 dimensional solenoidal manifolds and introduces new results on their structure and properties.

Findings

Topologically homogeneous, compact solenoidal manifolds are McCord solenoids.

These manifolds are principal Cantor-group bundles over compact manifolds.

They encode the commensurability properties of manifolds.

Abstract

In this paper we survey $n$-dimensional solenoidal manifolds for $n=1,2$ and 3, and present new results about them. Solenoidal manifolds of dimension $n$ are metric spaces locally modeled on the product of a Cantor set and an open $n$-dimensional disk. Therefore, they can be "laminated" (or "foliated") by $n$-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids i.e. are obtained as the inverse limit of an increasing tower of finite, regular covers of a compact manifold with an infinite and residually finite fundamental group. In this case their structure is very rich since they are principal Cantor-group bundles over a compact manifold and they behave like "laminated" versions of compact manifolds, thus they share many of their properties. These objects codify the commensurability properties of manifolds.