Consider the group and its subgroups and . is a canonical realization (up to an homeomorphism) of the complement of the trefoil knot , and is a canonical realization of the 6-fold branched cyclic cover of , which has 3-dimensional cohomology of 1-forms. Putting natural left-invariant Riemannian metrics on , it makes sense to ask which is the asymptotic homology performed by the Brownian motion in , 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 , 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 are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).
Cite this article
Jacques Franchi, Asymptotic windings over the trefoil knot. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 729–770DOI 10.4171/RMI/434