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 $\varepsilon$ such that every unitary representation of $G$ with a $(Q,\varepsilon )$-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 $G=\mathbb{Z}$, this shows that if $(n_{k})_{k\ge 1}$ is a sequence of integers such that $(e^{2i\pi \theta n_{k}})_{k\ge 1}$ is uniformly distributed in the unit circle for all real numbers $\theta $ except at most countably many, then $\{n_{k}\,;\,k\ge 1\}$ is a Kazhdan set in $\mathbb{Z}$ as soon as it generates $\mathbb{Z}$. 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 $\textrm{Aff}_{+}(\mathbb{R})$. This answers in particular a question from [B.~Bekka, P.~de la~Harpe, A.~Valette, Kazhdan's property (T), op. cit.].