Asymptotic windings over the trefoil knot

Abstract

Consider the group G:=PSL2(R) and its subgroups ?:= PSL2(Z) and ?':=DSL2(Z). G/? is a canonical realization (up to an homeomorphism) of the complement S3T of the trefoil knot T, and G/?' is a canonical realization of the 6-fold branched cyclic cover of S3T, which has a 3-dimensional cohomology of 1-forms. Putting natural left-invariant Riemannian metrics on G, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in G/?', describing thereby in an intrinsic way part of the asymptotic Brownian behavior in the fundamental group of the complement of the trefoil knot. A good basis of the cohomology of G/?', made of harmonic 1-forms, is calculated, and then the asymptotic Brownian behavior is obtained, by means of the joint asymptotic law of the integrals of the above basis along the Brownian paths. Finally the geodesics of G are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in G/?' is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).