Wild solenoids

Abstract

Dedicated to the memory of James T. Rogers, Jr.Abstract. A weak solenoid is a foliated space defined as the inverse limit of finite coverings of a closed compact manifold M . The monodromy of a weak solenoid defines an equicontinuous minimal action on a Cantor space X by the fundamental group G of M . The discriminant group of this action is an obstruction to this action being homogeneous. The discriminant vanishes if the group G 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 X. If the discriminant group never stabilizes as the diameter of the clopen set U 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 study of the properties of the discriminant group for Cantor actions. First, the tail equivalence class of the sequence of discriminant groups obtained for the restricted action on a neighborhood basis system of a point in X defines an invariant of the return equivalence class of the action, called the asymptotic discriminant, which is consequently an invariant of the homeomorphism class of the weak solenoid. Second, we construct uncountable collections of wild solenoids with pairwise distinct asymptotic discriminant invariants for a fixed base manifold M , and hence fixed finitely-presented group G, which are thus pairwise non-homeomorphic. The study in this work is the continuation of the seminal works on homeomorphisms of weak solenoids by Rogers and Tollefson in 1971, and is dedicated to the memory of Jim Rogers.

Arboreal Cantor actions

“…However, the notion of the Ellis group does not require finite generation, and finite generation was not used in any of the proofs in . One easily checks that Theorem and all results in which we use in this paper are true for countably generated groups as well. However, some related results on strong quasi‐analyticity in may require the finite (more precisely, compact) generation assumption.…”

“…One such invariant, called the asymptotic discriminant , has been introduced by the author joint with Hurder in , as a culmination of a series of papers on actions with non‐trivial isotropy groups joint with Dyer and Hurder . The asymptotic discriminant of a minimal equicontinuous action is an invariant of return equivalence of minimal Cantor systems.…”

“…That is, the cardinality of reflects the global properties of the action. The idea on how to use the Ellis group to develop a local invariant stems from and was fully developed in the joint paper with Hurder . The idea is as follows.…”

“…The asymptotic discriminant of an equicontinuous minimal group action was introduced and studied in and it was shown to be an invariant of return equivalence of group actions; see for a rigorous definition and properties of return equivalence. Again, even though makes an assumption of finite generation of the group , the proofs translate verbatim to the case of a countably generated . The introduction of the asymptotic discriminant allowed us to divide equicontinuous minimal Cantor actions into two large classes, as in the following definition.…”

“…An uncountable family of distinct wild actions of subgroups of for was constructed in . In , every finite group and every separable profinite group were realized as discriminant groups of stable actions of subgroups of , .…”