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.

References

The above convergence was conjectured by Buck for diffusion operators— that is, for polynomials with positive coefficients— and by Magee (personal communication) for arbitrary polynomials.

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

However, whether these representations a priori control the dynamics is unclear due to the problem that the monodromy groups need not be generating for all of $\Gamma$.

— Relative ($τ$), Expanders, and Decay of Correlations for certain Expanding Maps  (2609.05271 - Dougall, 4 Sep 2026) in Section 2, “Relative ($\tau$), non-generating subsets, and the inverse limit”