Asymptotic homology of the quotient of PSL₂() by a modular group

الأصل · EN

Consider G:= PSL₂()≡ T¹², a modular group Γ, and the homogeneous space Γ G ≡ T¹(Γ²). Endow G, and then Γ G, with a canonical left-invariant metric, thereby equipping it with a quasi hyperbolic geometry. Windings around handles and cusps of Γ G are calculated by integrals of closed 1-forms of Γ 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 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 Γ² (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.