Revista Matemática Iberoamericana

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Volume 21, Issue 3, 2005, pp. 729–770
DOI: 10.4171/RMI/434

Published online: 2005-12-31

Asymptotic windings over the trefoil knot

Jacques Franchi[1]

(1) Université de Strasbourg, France

Consider the group $G:=PSL_2(\mathbb R)$ and its subgroups $\Gamma:= PSL_2(\mathbb{Z})$ and $\Gamma':= DSL_2(\mathbb{Z})$. $G/\Gamma$ is a canonical realization (up to an homeomorphism) of the complement $\mathbb S^3\setminus T$ of the trefoil knot $T$, and $G/\Gamma'$ 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 $G/\Gamma'$, 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'$, 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 $G/\Gamma'$ is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).

Keywords: Trefoil knot, modular group, quasi-hyperbolic manifold, harmonic 1-forms, Brownian motion, geodesics, ergodic measures, geodesic flow, asymptotic laws

Franchi Jacques: Asymptotic windings over the trefoil knot. Rev. Mat. Iberoam. 21 (2005), 729-770. doi: 10.4171/RMI/434