The discriminant invariant of Cantor group actions

TL;DR

This paper introduces the discriminant invariant for Cantor group actions, linking it to the homogeneity of weak solenoids and providing new examples to illustrate subtle properties of these structures.

Contribution

It develops an alternative homogeneity criterion using the Ellis semigroup and explores the relationship between non-homogeneity and the discriminant invariant.

Findings

Discriminant invariant characterizes non-homogeneous weak solenoids.

New examples demonstrate subtle properties of group chains.

Alternative homogeneity condition via Ellis semigroup.

Abstract

In this work, we investigate the dynamical and geometric properties of weak solenoids, as part of the development of a "calculus of group chains" associated to Cantor minimal actions. The study of the properties of group chains was initiated in the works of McCord 1965 and Fokkink and Oversteegen 2002, to study the problem of determining which weak solenoids are homogeneous continua. We develop an alternative condition for the homogeneity in terms of the Ellis semigroup of the action, then investigate the relationship between non-homogeneity of a weak solenoid and its discriminant invariant, which we introduce in this work. A key part of our study is the construction of new examples that illustrate various subtle properties of group chains that correspond to geometric properties of non-homogeneous weak solenoids.