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Kazhdan sets in groups and equidistribution properties

Research paper by Catalin Badea, Sophie Grivaux

Indexed on: 16 Mar '16Published on: 16 Mar '16Published in: Mathematics - Group Theory



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 subset $Q$ of a group $G$ which generates $G$ 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 subset of $G$ which generates $G$ 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, $\{n_{k}\,;\,k\ge 1\}$ is a Kazhdan set in $\mathbb{Z}$ as soon as it generates $\mathbb{Z}$. This answers a question of Y. Shalom. We also obtain a complete characterization of Kazhdan sets in second countable locally compact abelian groups, in the Heisenberg groups, the $ax+b$ group, and $SL_2(\mathbb{R})$. This answers in particular a question from [B. Bekka, P. de la Harpe, A. Valette, Kazhdan's property (T), Cambridge Univ. Press, 2008]. Lastly, we provide some further applications of Kazhdan sets: we generalize a local rigidity result of Rapinchuk, and give a characterization of Kazhdan sets in terms of strong ergodicity of groups actions; this last result is related to a characterization of Property (T) due to Connes and Weiss.