Stabilized automorphism groups and full groups of odometers
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914243972956160 |
|---|---|
| author | Cortez, María Isabel Urria, Vicente |
| author_facet | Cortez, María Isabel Urria, Vicente |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stabilized automorphism groups and full groups of odometers Cortez, María Isabel Urria, Vicente Group Theory Dynamical Systems 37B05, 22C05 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. |
| title | Stabilized automorphism groups and full groups of odometers |
| topic | Group Theory Dynamical Systems 37B05, 22C05 |
| url | https://arxiv.org/abs/2508.20005 |