Expander property for congruence quotients of Γ+ over all moduli

Determine whether the family of congruence quotients of the semigroup Γ+, generated by the matrices A=[1 φ; 0 1], B=[1 0; φ 1], C=[φ φ; 1 φ], and D=[φ 1; φ φ] inside the (2,5,∞) Hecke triangle group, has the expander property uniformly for all moduli (including non-square-free moduli), i.e., whether the associated finite Cayley graphs of Γ+ modulo q form an expander family across all moduli.

Background

The authors discuss the potential applicability of the orbital circle method to their local-global problem but identify a key obstacle: the need for spectral expansion in congruence quotients of Γ+. While Γ is a lattice with Zariski closure SL2×SL2, Γ+ is a thin semigroup, and establishing uniform expansion over all moduli is not currently known.

This lack of a verified expander property (property τ or equivalent spectral gap) for all moduli—beyond the square-free case—is cited as a major obstruction to applying powerful analytic methods and is closely tied to the algebraic structure of the ambient group.