Orbit equivalence and classification of weak solenoids
arXiv:1803.02098
The paper investigates minimal equicontinuous actions that are locally quasi‑analytic, proving that continuous orbit equivalence implies return equivalence and that such actions are always locally quasi‑analytic for finitely‑generated virtually nilpotent groups, with applications to classifying nil‑solenoids via their topological full groups.
Abstract
In this work, we study minimal equicontinuous actions which are locally quasi-analytic. The first main result shows that for minimal equicontinuous actions which are locally quasi-analytic, continuous orbit equivalence of the actions implies return equivalence. This generalizes results of Cortez and Medynets, and of Li. The second main result is that if G is a finitely-generated, virtually nilpotent group, then every minimal equicontinuous action by G is locally quasi-analytic. As an application, we show that the homeomorphism type of a nil-solenoid is determined by the virtual topological full group of its monodromy action.
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