Disorder on the hyperbolic square lattice I: Anderson delocalization and absolutely continuous spectrum

We study the Anderson model on hyperbolic lattices. The graph is the Cartesian product of the Euclidean lattice with the regular hyperbolic square lattice having five squares at each vertex. We prove the existence of absolutely continuous spectrum and the absence of singular spectrum on a deterministic set of positive measure at weak disorder, and for $0<λ\le2$ when the single-site density is bounded below on $[-1,1]$. Truncated Cauchy distributions and controlled perturbations give purely absolutely continuous spectrum on intervals.