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 $M$.