Conjectures for strong convergence under PSL2(Z) action on projective lines over finite fields

Prove that, for fixed elements X_1,…,X_r in PSL2(Z), the permutation representations π_q induced by the Möbius action on P^1(F_q) satisfy strong convergence, as q→∞ over primes, of (π_q(X_1),…,π_q(X_r))|_{1^⊥} to (λ_{PSL2(Z)}(X_1),…,λ_{PSL2(Z)}(X_r)) in operator norm for all noncommutative polynomials P; in particular, establish this both for diffusion operators (polynomials with nonnegative coefficients) and for arbitrary polynomials, as conjectured by Buck and by Magee, respectively.

Background

The survey proposes a deterministic family built from the Möbius action of PSL2(Z) on the finite projective line, yielding explicit permutation representations π_q. The strong convergence question asks whether these representations approximate the regular representation in the sense of operator norms of noncommutative polynomials.

Buck conjectured the convergence for diffusion operators (positive-coefficient polynomials), and Magee conjectured the full version for arbitrary polynomials, linking deterministic models to the strong convergence paradigm.