Rigidity sequences, Kazhdan sets and group topologies on the integers

Catalin Badea (1) , Sophie Grivaux (1) , Étienne Matheron (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 LML - Laboratoire de Mathématiques de Lens

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.

Domaines: Mathématiques [math]; Systèmes dynamiques [math.DS]; Analyse fonctionnelle [math.FA]

Catalin Badea, Sophie Grivaux, Étienne Matheron. Rigidity sequences, Kazhdan sets and group topologies on the integers. Journal d'analyse mathématique, 2021, 143 (1), pp.313-347. ⟨10.1007/s11854-021-0165-4⟩. ⟨hal-02266204⟩