Asymptotic homology of the quotient of \(PSL_2(\BR)\) by a modular group

Abstract

Consider \( G:= PSL_2(\R)\equiv T^1\H^2\), a modular group \( \Gamma\), and the homogeneous space \( \Gamma\sm G \equiv T^1(\Gamma\sm\H^2)\). Endow \( G \), and then \( \Gamma\sm G \), with a canonical left-invariant metric, thereby equipping it with a quasi hyperbolic geometry. Windings around handles and cusps of \( \Gamma\sm G \) are calculated by integrals of closed 1-forms of \( \Gamma\sm G \). 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 \( \Gamma\sm G \) is responsible for a difference between the Brownian and geodesic asymptotic behaviours, difference which does not exist at the level of the Riemann surface \(\Gamma\sm\H^2\) (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.