Exact values of Kazhdan constants for some finite groups (Q1320193)

Let \(\Gamma\) be a group generated by a finite set \(S\) and let \(k_\Gamma(S)=\inf\{k_\Gamma(S,\pi):\pi\) is a unitary representation with no nontrivial common fixed points\} \[ k_\Gamma(S,\pi)=\inf_{|x|=1}\max_{s\in S}|\pi(s)x-x|. \] The main result of the paper is the exact computation of \(k_\Gamma(S)\) for \(\Gamma=\sigma_n\), the symmetric group of \(n\) objects, and \(S\) a special set of transpositions. There are applications to finite graphs.