  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.
