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It is well-known that the set of closed geodesics on a closed hyperbolic surface is dense. My questions: If this property still holds on finite-area hyperbolic surfaces, infinite-area hyperbolic ...

Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...