Spectral gap for constant-degree random Cayley graphs of finite simple groups

Determine whether constant-degree random Cayley graphs of finite simple groups form a family of expanders, equivalently whether they possess a spectral gap.

Background

The paper places sparse random circulant matrices in the broader setting of random Cayley graphs. It notes that random Cayley graphs are structured proxies for random graphs and that their expansion and spectral properties are important topics in the area.

Within this context, the paper explicitly identifies the expander question for constant-degree random Cayley graphs of finite simple groups as an important open problem. The formulation gives two equivalent descriptions: forming a family of expanders and possessing a spectral gap.