Asymptotic windings over the trefoil knot

Abstract

Consider the group $G:=PS{L}_2(\mathbb{R})$ and its subgroups $\Gamma :=PS{L}_2(\mathbb{Z})$ and ${\Gamma }^{\prime }:=DS{L}_2(\mathbb{Z})$. $G/\Gamma$ is a canonical realization (up to an homeomorphism) of the complement ${\mathbb{S}}^3\setminus T$ of the trefoil knot $T$, and $G/{\Gamma }^{\prime }$ is a canonical realization of the 6-fold branched cyclic cover of ${\mathbb{S}}^3\setminus T$, which has 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 $G/{\Gamma }^{\prime }$, 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 $G/{\Gamma }^{\prime }$, 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 $G/{\Gamma }^{\prime }$ is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).