Kazhdan constants and isomorphic graph pairs

Abstract

Let \(G\) be a finite group, and let \(\Gamma\) be a~subset of \(G\). The Kazhdan constant of the pair \((G,\Gamma)\) is defined to be the maximum distance we can guarantee that an arbitrary unit vector in an arbitrary nontrivial irreducible unitary representation space of \(G\) can be moved by some element of \(\Gamma\). The Kazhdan constant relates to the expansion properties of the Cayley graph generated by \(G\) and \(\Gamma\), and has been much studied in this context. Different pairs \((G_1,\Gamma_1)\) and \((G_2,\Gamma_2)\) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To what extent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we construct an infinite family of such pairs where the Kazhdan constants are unequal. Other relevant results are presented as well.