# Stabilized automorphism groups and full groups of odometers

## TL;DR

This paper explores the relationship between stabilized automorphism groups and full groups of odometers, revealing conditions under which these groups are isomorphic and their implications for orbit equivalence.

## Contribution

It establishes that stabilized automorphism groups of free exact odometers coincide with topological full groups and characterizes their isomorphism conditions.

## Key findings

- Stabilized automorphism groups equal topological full groups for certain odometers.
- Isomorphism of stabilized automorphism groups characterizes orbit equivalence in specific cases.
- Continuous orbit equivalence implies isomorphic stabilized automorphism groups.

## Abstract

In this article, we show that the stabilized automorphism group of free exact odometers arising from actions of finitely generated residually finite groups coincides with the topological full group of the odometer acting on itself by right multiplication. We then prove that two free exact odometers have isomorphic stabilized automorphism groups if and only if they have isomorphic clopen subgroups of the same index. As a consequence, continuous orbit equivalence implies isomorphic stabilized automorphism groups, while for free $\mathbb{Z}^d$-odometers, isomorphic stabilized automorphism groups imply orbit equivalence. In general, neither continuous orbit equivalence nor orbit equivalence is equivalent to having isomorphic stabilized automorphism groups.