# Rigidity sequences, Kazhdan sets and group topologies on the integers

## TL;DR

This paper explores the relationships among rigidity, Kazhdan, and nullpotent sequences of integers, establishing new implications and criteria, and providing a novel proof of the density of rigidity sequences in the Bohr topology.

## Contribution

It demonstrates that rigidity sequences are non-Kazhdan and nullpotent, introduces probabilistic and Baire category methods for characterizing rigidity sequences, and offers a new proof of their density in the Bohr topology.

## Key findings

- Rigidity sequences are non-Kazhdan and nullpotent.
- Existence of sequences that are both nullpotent and Kazhdan.
- Rigidity sequences can be dense in the Bohr topology.

## 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 $\mathbb{Z}$ for the Bohr topology, a result originally due to Griesmer.