Strong uniform expansion in $\mathrm{SL}(2,p)$

TL;DR

This paper proves that for a large set of primes, all Cayley graphs of the groups SL(2,p) form a family of expander graphs, highlighting uniform expansion properties in these groups.

Contribution

It establishes the existence of an infinite set of primes with density one for which all Cayley graphs of SL(2,p) are expanders, a significant advance in group expansion theory.

Findings

Infinite set of primes with density one identified

All Cayley graphs of SL(2,p) are expanders for these primes

Supports uniform expansion properties in SL(2,p) groups

Abstract

We show that there is an infinite set of primes $\mathcal{P}$ of density one, such that the family of \textit{all} Cayley graphs of $\mathrm{SL}(2,p)$%, $p\in \mathcal{P}$, is a family of expanders.