# Asymptotic windings over the trefoil knot.

Revista Matemática Iberoamericana (2005)

- Volume: 21, Issue: 3, page 729-770
- ISSN: 0213-2230

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topFranchi, Jacques. "Asymptotic windings over the trefoil knot.." Revista Matemática Iberoamericana 21.3 (2005): 729-770. <http://eudml.org/doc/41949>.

@article{Franchi2005,

abstract = {Consider the group G:=PSL2(R) and its subgroups Γ:= PSL2(Z) and Γ':=DSL2(Z). G/Γ is a canonical realization (up to an homeomorphism) of the complement S3T of the trefoil knot T, and G/Γ' 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/Γ', 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/Γ', 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/Γ' is calculated, by reduction to its Brownian analogue, though it is not precisely the same (counter to the hyperbolic case).},

author = {Franchi, Jacques},

journal = {Revista Matemática Iberoamericana},

keywords = {Nudos topológicos; Movimiento browniano; trefoil knot; quasi-hyperbolic manifold; harmonic forms; Brownian motion; geodesic flow; ergodic measures; asymptotic laws},

language = {eng},

number = {3},

pages = {729-770},

title = {Asymptotic windings over the trefoil knot.},

url = {http://eudml.org/doc/41949},

volume = {21},

year = {2005},

}

TY - JOUR

AU - Franchi, Jacques

TI - Asymptotic windings over the trefoil knot.

JO - Revista Matemática Iberoamericana

PY - 2005

VL - 21

IS - 3

SP - 729

EP - 770

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

LA - eng

KW - Nudos topológicos; Movimiento browniano; trefoil knot; quasi-hyperbolic manifold; harmonic forms; Brownian motion; geodesic flow; ergodic measures; asymptotic laws

UR - http://eudml.org/doc/41949

ER -