JournalsrmiVol. 21, No. 3pp. 729–770

Asymptotic windings over the trefoil knot

  • Jacques Franchi

    Université de Strasbourg, France
Asymptotic windings over the trefoil knot cover
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Abstract

Consider the group G:=PSL2(R)G:=PSL_2(\mathbb R) and its subgroups Γ:=PSL2(Z)\Gamma:= PSL_2(\mathbb{Z}) and Γ:=DSL2(Z)\Gamma':= DSL_2(\mathbb{Z}). G/ΓG/\Gamma is a canonical realization (up to an homeomorphism) of the complement S3T\mathbb S^3\setminus T of the trefoil knot TT, and G/ΓG/\Gamma' is a canonical realization of the 6-fold branched cyclic cover of S3T\mathbb S^3\setminus T, which has 3-dimensional cohomology of 1-forms. Putting natural left-invariant Riemannian metrics on GG, it makes sense to ask which is the asymptotic homology performed by the Brownian motion in G/ΓG/\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/ΓG/\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 GG are determined, a natural class of ergodic measures for the geodesic flow is exhibited, and the asymptotic geodesic behavior in G/ΓG/\Gamma' 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. 729–770

DOI 10.4171/RMI/434