Type invariants for non-abelian odometers

TL;DR

This paper introduces the type and typeset invariants for equicontinuous group actions on Cantor sets, providing tools to classify non-abelian odometers and related solenoidal manifolds.

Contribution

It defines new invariants for generalized odometers, proves their invariance under return equivalence, and applies them to classify solenoidal manifolds.

Findings

Type is an invariant of return equivalence.

Typesets are commensurable for return equivalent actions.

Invariants help distinguish non-abelian odometers and classify solenoidal manifolds.

Abstract

In this work, we introduce the type and typeset invariants for equicontinuous group actions on Cantor sets; that is, for generalized odometers. These invariants are collections of equivalence classes of asymptotic Steinitz numbers associated to the action. We show the type is an invariant of the return equivalence class of the action. We introduce the notion of commensurable typesets, and show that two actions which are return equivalent have commensurable typesets. Examples are given to illustrate the properties of the type and typeset invariants. The type and typeset invariants are used to define homeomorphism invariants for solenoidal manifolds.