Deterministic strong convergence from Ramanujan constructions (PSL2(F_q))

Determine whether the matrices obtained by applying the regular representation of PSL2(F_q) to the explicit Lubotzky–Phillips–Sarnak/Margulis generators converge strongly, as q→∞ over primes, to the regular representation of the free group’s generators; equivalently, establish deterministic strong convergence for these explicit Cayley graph constructions.

Background

Explicit Ramanujan graph constructions provide optimal spectral gaps for specific Cayley graphs of PSL2(F_q), prompting the natural question of whether a deterministic analogue of strong convergence holds in this setting.

Voiculescu raised this deterministic strong convergence question in connection with these constructions. Despite decades of progress, the problem remains open and current techniques used to prove the Ramanujan property do not seem to address strong convergence.