Comments on: Filling geodesics and hyperbolic complements

By: model railroad

model railroad — Mon, 19 Nov 2012 16:14:06 +0000

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By: Characteristic classes of foliations « Geometry and the imagination

Characteristic classes of foliations « Geometry and the imagination — Tue, 21 Feb 2012 12:45:30 +0000

[...] . For example, if , the unit tangent bundle of a hyperbolic surface with its stable foliation (see this post), then is the geodesic flow itself (up to a change of orientation), and the Godbillon-Vey [...]

By: Danny Calegari

Danny Calegari — Sat, 11 Feb 2012 22:36:23 +0000

Hi Ian – thanks for the link to Porti’s paper; I hadn’t seen it before. And thanks for making the connection with Montesinos links. I don’t have any immediate ideas about regeneration in this context, but maybe something will occur to me after I look at Porti’s paper.

By: Ian Agol

Ian Agol — Sat, 11 Feb 2012 22:18:28 +0000

I suppose this theorem was known for 2-fold covers of certain hyperbolic Montesinos links, which have geodesics in the unit tangent bundle of a surface (orbifold) which are invariant under an involution. This might indicate the overlap with alternating knots, since some Montesinos links are alternating. I think that your result also extends to unit tangent bundles of orbifolds. Porti has proven that the hyperbolic metric may be “regenerated” from the hyperbolic surface: http://front.math.ucdavis.edu/1003.2494

I wonder if there’s a similar regeneration in the context you’re considering?