The prime spectrum of solenoidal manifolds

TL;DR

This paper introduces new invariants called asymptotic Steinitz orders and prime spectra for solenoidal manifolds, proving they are homeomorphism invariants and exploring their properties in minimal equicontinuous Cantor actions.

Contribution

It defines and studies the invariants of asymptotic Steinitz orders and prime spectra for solenoidal manifolds and their associated Cantor actions, including invariance and stability results.

Findings

Prime spectra are invariants under homeomorphism.

Stable nilpotent Cantor actions can have arbitrary prime spectra.

First examples of wild nilpotent Cantor actions are provided.

Abstract

A solenoidal manifold is the inverse limit space of a tower of proper coverings of a compact manifold. In this work, we introduce new invariants for solenoidal manifolds, their asymptotic Steinitz orders and their prime spectra, and show they are invariants of the homeomorphism type. These invariants are formulated in terms of the monodromy Cantor action associated to a solenoidal manifold. To this end, we continue our study of invariants for minimal equicontinuous Cantor actions. We introduce the three types of prime spectra associated to such actions, and study their invariance properties under return equivalence. As an application, we show that a nilpotent Cantor action with finite prime spectrum must be stable. Examples of stable actions of the integer Heisenberg group are given with arbitrary prime spectrum. We also give the first examples of nilpotent Cantor actions which are wild.