2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119456Consider $ G:= PSL_2(\R)\equiv T^1\H^2$, a modular group $ Γ$, and the homogeneous space $ Γ\sm G \equiv T^1(Γ\sm\H^2)$. Endow $ G $, and then $ Γ\sm G $, with a canonical left-invariant metric, thereby equipping it with a quasi hyperbolic geometry. Windings around handles and cusps of $ Γ\sm G $ are calculated by integrals of closed 1-forms of $ Γ\sm G $. The main results express, in both Brownian and geodesic cases, the joint convergence of the law of these integrals, with a stress on the asymptotic independence between slow and fast windings. The non-hyperbolicity of $ Γ\sm G $ is responsible for a difference between the Brownian and geodesic asymptotic behaviours, difference which does not exist at the level of the Riemann surface $Γ\sm\H^2$ (and generally in hyperbolic cases). Identification of the cohomology classes of closed 1-forms and with harmonic 1-forms, and equidistribution of large geodesic spheres, are also addressed.36 pagesProbabilityprimary : 58J65 ; secondary : 60J65, 37D40, 37D30, 37A50, 20H05, 53C22text