Stabilized automorphism groups and full groups of odometers

TL;DR

The paper proves that for free exact odometers arising from residually finite groups, the stabilized automorphism group ${\rm Aut}^{\infty}(X,\alpha_X,G)$ coincides with the topological full group of the right odometer $[[R_X]]$, and it characterizes isomorphisms between stabilized groups by the existence of isomorphic clopen subgroups of the same finite index. It then links these algebraic invariants to orbit equivalence notions, showing that continuous orbit equivalence implies isomorphism of stabilized groups, while the converse fails in general; for $\mathbb{Z}^d$-odometers, stabilized-group isomorphism does imply orbit equivalence. The paper also provides a number of sharp counterexamples in the $\mathbb{Z}^2$-case illustrating that stabilized automorphism groups are a nuanced invariant: they do not determine continuous orbit equivalence, nor are they determined by orbit equivalence, and they may fail to reflect the topological group structure.\

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.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 3.1
• proof
• Remark 3.2