Compact aspherical solenoids

TL;DR

This paper classifies maps between compact aspherical solenoids, including p-adic and hyperbolic types, using shape theory, and establishes a Dehn-Nielsen-type theorem for their self-homotopy equivalences.

Contribution

It provides a classification of maps and self-homotopy equivalences of compact aspherical solenoids, extending previous results to a broader class of these spaces.

Findings

Classification of maps between aspherical solenoids up to homotopy

A Dehn-Nielsen-type theorem for self-homotopy equivalences

Extension of known results to universal hyperbolic solenoid

Abstract

We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes $P$-adic solenoids, as well as the universal hyperbolic solenoid of Teichm\"{u}ller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove a Dehn-Nielsen-type theorem for self-homotopy equivalences of such a solenoid. This generalizes a result of Odden regarding the universal hyperbolic solenoid.