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Rigidity sequences, Kazhdan sets and group topologies on the integers

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Abstract

We study the relationships between three different classes of sequences (or sets) of integers, namely rigidity sequences, Kazhdan sequences (or sets) and nullpotent sequences. We prove that rigidity sequences are non-Kazhdan and nullpotent, and that all other implications are false. In particular, we show by probabilistic means that there exist sequences of integers which are both nullpotent and Kazhdan. Moreover, using Baire category methods, we provide general criteria for a sequence of integers to be a rigidity sequence. Finally, we give a new proof of the existence of rigidity sequences which are dense in ℤ for the Bohr topology, a result originally due to Griesmer.

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Correspondence to Sophie Grivaux.

Additional information

This work was supported in part by the project FRONT of the French National Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI (ANR-11-LABX-0007-01).

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Badea, C., Grivaux, S. & Matheron, É. Rigidity sequences, Kazhdan sets and group topologies on the integers. JAMA 143, 313–347 (2021). https://doi.org/10.1007/s11854-021-0165-4

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  • DOI: https://doi.org/10.1007/s11854-021-0165-4

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