Isometry groups of infinite-genus hyperbolic surfaces

Given a 2-manifold, fundamental question to ask is which groups can be realized as the isometry group of Riemannian metric constant curvature on manifold. In this paper, we give nearly complete classification such for infinite-genus 2-manifolds with no planar ends. Surprisingly, show there an uncountable class where every countable (namely, those self-similar end spaces). We apply result obtain obstructions standard theoretic properties homeomorphisms, diffeomorphisms, and mapping 2-manifolds. For example, none these satisfy Tits Alternative; are coherent; linear; cyclically or linearly orderable; residually finite. As second application, algebraic rigidity groups.