Strong convergence for higher-dimensional hyperbolic manifolds via permutation representations

Determine whether the fundamental group Γ of a closed hyperbolic manifold of dimension d≥3 admits a sequence of random permutation representations that converges strongly to the regular representation λ_Γ, thereby enabling random cover constructions with new eigenvalues asymptotically above the universal bound.

Background

The survey explains that Hide–Magee’s method for surfaces can be reduced to strong convergence of permutation representations, enabling sharp spectral results for random covers of hyperbolic surfaces.

In higher dimensions, universal upper bounds and parts of the method extend, but the key missing component is whether fundamental groups of such manifolds admit strongly convergent permutation representation sequences.