Kazhdan sets in groups and equidistribution properties

TL;DR

This paper introduces a new criterion for identifying Kazhdan sets in various groups using harmonic analysis, and applies it to characterize such sets in groups like integers, Heisenberg, and affine groups.

Contribution

It provides a novel criterion for Kazhdan sets based on representation invariance and applies it to characterize these sets in multiple classes of groups.

Findings

A new criterion for Kazhdan sets using unitary representations.

Characterization of Kazhdan sets in abelian, Heisenberg, and affine groups.

Answering open questions about equidistribution and Kazhdan sets.

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.].