Kazhdan sets in groups and equidistribution properties
Résumé
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 $G=Z$ is the group of integers, 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 $Z$ 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 $\textrm{Aff}_{+}(R)$.This answers in particular a question from [B.~Bekka, P.~de la~Harpe, A.~Valette, Kazhdan's property (T), op. cit.].
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