Kazhdan sets in groups and equidistribution properties

Abstract

Using functional and harmonic analysis methods, we study Kazhdan sets in topological groups which do not necessarily have Property (T). We provide a new criterion for a generating subset Q of a group G to be a Kazhdan set; it relies on the existence of a positive number ε such that every unitary representation of G with a (Q,ε)-invariant vector has a finite dimensional subrepresentation. Using this result, we give an equidistribution criterion for a generating subset of G to be a Kazhdan set. In the case where G=Z, this shows that if (nk)k≥1 is a sequence of integers such that (e2iπθnk)k≥1 is uniformly distributed in the unit circle for all real numbers θ except at most countably many, then {nk;k≥1} is a Kazhdan set in Z as soon as it generates Z. This answers a question of Y. Shalom from [B. Bekka, P. de la Harpe, A. Valette, Kazhdan's property (T), Cambridge Univ. Press, 2008]. We also obtain characterizations of Kazhdan sets in second countable locally compact abelian groups, in the Heisenberg groups and in the group Aff+(R). This answers in particular a question from [B. Bekka, P. de la Harpe, A. Valette, Kazhdan's property (T), op. cit.].