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.

References

This question was raised by Voiculescu p.\ 146 in an early paper that motivated the development of strong convergence of random matrices by Haagerup and Thorbj{\o}rnsen. However, the deterministic question remains open, and the methods of appear to be powerless for addressing this question.

— The strong convergence phenomenon  (2507.00346 - Handel, 1 Jul 2025) in Section 6.4 (Deterministic constructions)

However, it remains not clear, whether a closed form or algebraic formula exists.

— Superadditivity of classical communication over quantum channels via random and deterministic permutations  (2608.25961 - Lovitz et al., 26 Aug 2026) in Section 1, subsection “Discussions and organization,” second bullet