Kazhdan sets in groups and equidistribution properties

摘要

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 epsilon such that every unitary representation of G with a (Q, epsilon)-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 (n(k))(k >= 1) is a sequence of integers such that (e(2i pi theta nk))(k >= 1) is uniformly distributed in the unit circle for all real numbers theta except at most countably many, then {n(k); 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.]. (C) 2017 Elsevier Inc. All rights reserved.