Wild solenoids
Steven Hurder, Olga Lukina

TL;DR
This paper investigates the properties of weak solenoids, focusing on the discriminant group of their associated Cantor actions, and introduces the asymptotic discriminant as an invariant to distinguish non-homeomorphic wild solenoids.
Contribution
It defines the asymptotic discriminant as an invariant of weak solenoids and constructs uncountably many wild solenoids with distinct invariants, advancing classification methods.
Findings
The tail equivalence class of discriminant groups is an invariant of the homeomorphism class.
Uncountably many wild solenoids have pairwise distinct asymptotic discriminants.
The asymptotic discriminant is an invariant of the return equivalence class of the action.
Abstract
A weak solenoid is a foliated space defined as the inverse limit of finite coverings of a closed compact manifold . The monodromy of a weak solenoid defines an equicontinuous minimal action on a Cantor space by the fundamental group of . The discriminant group of this action is an obstruction to this action being homogeneous. The discriminant vanishes if the group is abelian, but there are examples of actions of nilpotent groups for which the discriminant is non-trivial. The action is said to be stable if the discriminant group remains unchanged for the induced action on sufficiently small clopen neighborhoods in . If the discriminant group never stabilizes as the diameter of the clopen set tends to zero, then we say that the action is unstable, and the weak solenoid which defines it is said to be wild. In this work, we show two main results in the course of our…
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Topological and Geometric Data Analysis
