Asympotic windings over the trefoil knot

• Autores: J. Franchi
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 21, Nº 3, 2005, págs. 729-770
• Idioma: inglés
• DOI: 10.4171/rmi/434
• Títulos paralelos:
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• Resumen

  • Consider the group G:=PSL2(R) and its subgroups ?:= PSL2(Z) and ?':=DSL2(Z). G/? is a canonical realization (up to an homeomorphism) of the complement S3T of the trefoil knot T, and G/?' is a canonical realization of the 6-fold branched cyclic cover of S3T, which has a 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/?', 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/?', 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/?' is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).