# Orbit equivalence and classification of weak solenoids

## TL;DR

This paper investigates the relationship between orbit equivalence and return equivalence in minimal equicontinuous actions, establishing new conditions under which these notions coincide and applying results to classify nil-solenoids.

## Contribution

It proves that for locally quasi-analytic minimal equicontinuous actions, orbit equivalence implies return equivalence, and shows all such actions by finitely-generated virtually nilpotent groups are locally quasi-analytic.

## Key findings

- Orbit equivalence implies return equivalence in locally quasi-analytic actions.
- All minimal equicontinuous actions by finitely-generated virtually nilpotent groups are locally quasi-analytic.
- The homeomorphism type of a nil-solenoid is determined by the virtual topological full group.

## 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.