Asymptotic windings over the trefoil knot

Abstract

Consider the group G := P SL 2 (R) and its subgroups Γ := P SL 2 (Z) and Γ := DSL 2 (Z). G/Γ is a canonical realization (up to an homeomorphism) of the complement S 3 \ T of the trefoil knot T , and G/Γ is a canonical realization of the 6-fold branched cyclic cover of S 3 \ 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/Γ , 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/Γ , 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/Γ is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).

A Central Limit Theorem for the Kontsevich-Zorich Cocycle

“…See also [Fra05, Lemma 8] for a short potential theoretic proof (using Doob's h-process) of this fact. The approach to proving the CLT in [Fra05], with the aid of a stopping time, is what we will essentially follow in the sequel, though in our case the proof here is simpler, in view of the Lipschitz property of the Kontsevich-Zorich cocycle and Ancona's estimate.…”