Strong convergence for a hyperbolic 3-manifold relation (Gieseking group) under uniform permutation sampling

Determine whether uniformly sampling permutation matrices U_1^N,U_2^N conditioned on the relation (U_1^N)^2 (U_2^N)^2 = U_2^N U_1^N yields a sequence of permutation representations of the group Γ = ⟨g_1,g_2 | g_1^2 g_2^2 = g_2 g_1⟩ that converges strongly to the left-regular representation λ_Γ; that is, prove strong convergence for this dependent model of random permutation matrices.

Background

The authors survey strong convergence beyond free groups by considering random models for non-free groups arising as fundamental groups of manifolds. For the Gieseking manifold group, the natural random model involves sampling permutation matrices subject to a nontrivial relation, leading to a dependent ensemble.

While strong convergence has been proved for several non-free settings (e.g., surface groups via specialized models), the analogous question for this simple 3-manifold group remains unresolved.