  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.
