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December, 2005 Asymptotic windings over the trefoil knot
Jacques Franchi
Rev. Mat. Iberoamericana 21(3): 729-770 (December, 2005).


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).


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Jacques Franchi. "Asymptotic windings over the trefoil knot." Rev. Mat. Iberoamericana 21 (3) 729 - 770, December, 2005.


Published: December, 2005
First available in Project Euclid: 11 January 2006

zbMATH: 1115.58029
MathSciNet: MR2231009

Primary: 58J65
Secondary: 20H05 , 37A50 , 37D30 , 37D40 , 53C22 , 60J65

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

Rights: Copyright © 2005 Departamento de Matemáticas, Universidad Autónoma de Madrid


Vol.21 • No. 3 • December, 2005
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