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We study the closure of horocycles on rank 1 nonpositively curved surfaceswith finitely generated fundamental group. Each horocycle is closed or dense ona certain subset of the unit tangent bundle. In fact, we classify eachhalf-horocycle in terms of the associated geodesic rays. We also determine thenonwandering set of the horocyclic flow and characterize the surfaces admittinga minimal set for this flow.

Topology of horocycles on geometrically finite nonpositively curved surfaces