2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66603We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these equivalences.To appear in Geometriae Dedicata, 23 pages, 8 figures, see also http://www.math.columbia.edu/~cleinGeometric Topology57M50; 51M25text