Fourier rearrangements on discrete groups: L^p and operator-norm subsequence convergence

Abstract

Let Γ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For f belonging to the reduced group C^* algebra C_r^*(Γ), we construct a rearrangement of its individual Fourier terms with a subsequence converging to f in the operator norm. For every 2≤ p<∞ and f belonging to the noncommutative L^p space $ L^p(\hat Γ)$, we obtain convergence in the L^p norm. The proof combines uniformly bounded finite-support Fourier cutoffs with an elementary Bernoulli even-moment estimate. We give a self-contained free-group argument, then establish a general criterion and verify it for the two classes above. The operator-norm conclusion establishes the Révész property for these groups. In particular, it confirms the polynomial-growth conjecture of Hamm, Hayes, and Petrosyan and disproves their conjecture that the free group F does not have this property.

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BibTeX RIS

BibTeX

@misc{mei2026fourier,
  title = {{Fourier rearrangements on discrete groups: $L^p$ and operator-norm subsequence convergence}},
  author = {Tao Mei and Sebastian Vargas-Loaiza},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15823},
  eprint = {2609.15823},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15823},
}

RIS

TY  - UNPB
TI  - Fourier rearrangements on discrete groups: L^p and operator-norm subsequence convergence
AU  - Tao Mei
AU  - Sebastian Vargas-Loaiza
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15823
UR  - https://arxiv.org/abs/2609.15823
AB  - Let Γ be an infinite finitely generated group that is hyperbolic or of polynomial growth. For f belonging to the reduced group C^* algebra C_r^*(Γ), we construct a rearrangement of its individual Fourier terms with a subsequence converging to f in the operator norm. For every 2≤ p<∞ and f belonging to the noncommutative L^p space $ L^p(\hat Γ)$, we obtain convergence in the L^p norm. The proof combines uniformly bounded finite-support Fourier cutoffs with an elementary Bernoulli even-moment estimate. We give a self-contained free-group argument, then establish a general criterion and verify it for the two classes above. The operator-norm conclusion establishes the Révész property for these groups. In particular, it confirms the polynomial-growth conjecture of Hamm, Hayes, and Petrosyan and disproves their conjecture that the free group F does not have this property.
ER  -

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