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arXiv · 2610.06488

On the centralizer and normalizer groups of odometers and Toeplitz subshifts

Jaime Gómez·Samuel Petite
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Abstract

For residually finite groups, we study the automorphism and normalizer groups of odometers and their symbolic extensions: Toeplitz subshifts. In relation to orbit equivalence theory, we show that two continuous orbit equivalent $G$-odometers have commensurable normalizers when $G=\mathbb{Z}^d$, but that this result fails for other nilpotent groups $G$. The automorphism and normalizer groups of any Toeplitz subshift embed into those of its underlying odometer; despite this constraint, they exhibit considerable flexibility. For instance, we exhibit examples of low-complexity $G$-Toeplitz subshifts whose centralizer is isomorphic to $G$ (or, at the other extreme, restricted to its center) yet whose normalizer group is as large as possible.

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Jaime Gómez, Samuel Petite. 2026-10-05. On the centralizer and normalizer groups of odometers and Toeplitz subshifts. https://arxiv.org/abs/2610.06488

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