Random Schreier graphs and expanders

TL;DR

This paper demonstrates that random Schreier graphs with logarithmic size exhibit spectral gaps with high probability, extending known results for Cayley graphs and analyzing spectral properties for nilpotent groups.

Contribution

It extends the Alon-Roichman theorem to Schreier graphs and provides bounds on the number of generators needed for spectral gaps depending on the group action.

Findings

Random Schreier graphs with $C \, ext{log} |\, ext{Omega}|$ elements have high probability of spectral gap.

Spectral gap bounds depend on the specific group action on the set.

Estimates of spectral gaps are provided for nilpotent groups.

Abstract

Let the group $G$ act transitively on the finite set $\Omega$, and let $S \subseteq G$ be closed under taking inverses. The Schreier graph $Sch(G \circlearrowleft \Omega,S)$ is the graph with vertex set $\Omega$ and edge set $\{ (\omega,\omega^s) : \omega \in \Omega, s \in S \}$. In this paper, we show that random Schreier graphs on $C \log|\Omega|$ elements exhibit a (two-sided) spectral gap with high probability, magnifying a well known theorem of Alon and Roichman for Cayley graphs. On the other hand, depending on the particular action of $G$ on $\Omega$, we give a lower bound on the number of elements which are necessary to provide a spectral gap. We use this method to estimate the spectral gap when $G$ is nilpotent.