Spectral gap for Pollicott-Ruelle resonances on random coverings of Anosov surfaces

Let $(M,g)$ be a closed Riemannian surface with Anosov geodesic flow. We prove the existence of a spectral gap for Pollicott–Ruelle resonances on random finite coverings of $M$ in the limit of large degree, which is expected to be optimal. The proof combines the recent strong convergence results of Magee, Puder and van Handel for permutation representations of surface groups with an analysis of the spherical mean operator on the universal cover of $M$.

💡 Research Summary

The paper investigates the spectral properties of Pollicott–Ruelle resonances associated with the geodesic flow on a closed Riemannian surface (M,g) whose flow is Anosov. The main goal is to prove that, for random finite-degree coverings of M with degree n tending to infinity, there exists a uniform resonance‑free strip to the right of a certain vertical line in the complex plane. This “spectral gap” is shown to be optimal in the sense that its width coincides with the quantity δ₀ = ½ Pr(−2ψ_u), where ψ_u is the unstable Jacobian and Pr denotes topological pressure.

The authors first recall the functional‑analytic framework for Anosov flows: the resolvent (X+z)⁻¹ initially defined for Re z>0 extends meromorphically to the whole complex plane, and its poles are the Pollicott–Ruelle resonances. Existing results (Butterley–Livernani, Faure–Sjöstrand) guarantee a finite number of resonances in the half‑plane Re z>−γ₀/2+ε, where γ₀ is the minimal exponential growth rate of the unstable Jacobian. However, these results do not control the location of resonances for families of coverings.

The novelty of the work lies in combining two recent advances. First, Magee, Puder and van Handel proved that for a surface group Γ, a uniformly random homomorphism φₙ: Γ→Sₙ produces a permutation representation that strongly converges to the regular representation λ_Γ as n→∞. Strong convergence means that for any finite linear combination w of group elements, the operator norms ‖ρₙ(w)−λ_Γ(w)‖ tend to zero. This probabilistic statement holds asymptotically almost surely (a.a.s.).

Second, the authors develop a low‑frequency propagation analysis on the universal cover \tilde M. By studying the spherical mean operator on \tilde M, they obtain a quantitative estimate showing that the resolvent of the lifted vector field X has no poles in the region Re z>δ₀, uniformly over all flat unitary bundles. This analysis hinges on the pressure formula \