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PreprintCC BY 4.0arXiv

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=ℤd, 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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Cite

BibTeXRIS
BibTeX
@misc{gomez2026centralizer,
  title = {{On the centralizer and normalizer groups of odometers and Toeplitz subshifts}},
  author = {Jaime Gómez and Samuel Petite},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06488},
  eprint = {2610.06488},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06488},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the centralizer and normalizer groups of odometers and Toeplitz subshifts
AU  - Jaime Gómez
AU  - Samuel Petite
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06488
UR  - https://arxiv.org/abs/2610.06488
AB  - 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=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.
ER  -