Rigidity sequences, Kazhdan sets and group topologies on the integers

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.

Rigidity, weak mixing, and recurrence in abelian groups

“…We now deduce topological corollaries from Theorem A, including an extension of [3,Theorem 3.8]. First, we need two definitions from [4].…”

“…In [4], it is shown that a sequence (n k ) k∈N in Z is a TB-sequence if and only if λ n k → 1 for infinitely many λ ∈ T. By Lemma 2.7, this means that TB-sequences in Z are rigidity sequences for ergodic systems with discrete spectrum. Badea, Grivaux, and Matheron use this observation to conclude that TB-sequences are rigidity sequences (see [3,Theorem 3.8]).…”

“…More generally, following Ruzsa [36], we say that a sequence (a n ) n∈N is nullpotent if there is some Hausdorff group topology τ on Γ such that a n → 0 in (Γ, τ ). Such sequences are also called T-sequences by Protasov and Zelenyuk [34], but we stick to Ruzsa's terminology to be consistent with [3]. Badea, Grivaux, and Matheron showed that every rigidity sequence in Z is nullpotent (see [3,Corollary 3.6]).…”

“…Such sequences are also called T-sequences by Protasov and Zelenyuk [34], but we stick to Ruzsa's terminology to be consistent with [3]. Badea, Grivaux, and Matheron showed that every rigidity sequence in Z is nullpotent (see [3,Corollary 3.6]). We extend this to our setting under the assumption that the measure-preserving action is free (as discussed in the introduction, this assumption is extraneous in Z).…”

“…Though it is not stated explicitly, Corollary 1.3 also follows from the work of Badea, Grivaux, and Matheron in [3]. In particular, they establish, for every discrete abelian group, the existence of a rigidity sequence that is dense in the Bohr topology (see [3,Section 5b.2]). This is a weaker result than Theorem C but sufficient to prove Corollary 1.3.…”