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Comment on Topologically non-trivial highways by Ars Mathematica » Blog Archive » Topologically Non-Trivial Highway

Ars Mathematica » Blog Archive » Topologically Non-Trivial Highway — Thu, 17 Jul 2008 19:23:48 +0000

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Comment on Rank vs. Genus by Jesse Johnson

Jesse Johnson — Mon, 14 Jul 2008 14:37:54 +0000

Thanks for pointing that out. The updated version of their paper can be found here.

Comment on Rank vs. Genus by Henry Wilton

Henry Wilton — Sun, 13 Jul 2008 16:49:35 +0000

There is a significant residue from the Abert–Nikolov paper you’re referring to. Despite the problems with the Dooley–Golodets paper they relied on, they have nevertheless shown that the Rank vs Heegard genus problem is connected to the Fixed Price problem in dynamics. The third version of their paper on the arXiv makes this clear.

Comment on Mapping classes vs. Homeomorphisms by JL

JL — Thu, 10 Jul 2008 03:05:44 +0000

That is not quite an analogous identification in higher dimensions, because an orbit of a (marked) ideal simplex (e.g. a regular one) does not take up the space of ideal simplices; in three dimensions the former has dimension 6 and the latter 8. Is there any meaningful interpretation of the set of ordered quadruple points of S^2?

Comment on Mapping classes vs. Homeomorphisms by Ian Agol

Ian Agol — Thu, 10 Jul 2008 01:34:10 +0000

Andy’s comment is the sort of thing I had in mind, although there’s not really a canonical identification. As he points out, I should have said ideal triangles with the vertices marked. The orientation preserving isometries of the hyperbolic plane act freely on the set of such marked triangles. But one may also identify the orientation preserving isometries with the unit tangent bundle in two dimensions, since an isometry is determined by where it takes a unit vector. The corresponding identification in higher dimensions is to take an orbit of a (marked) ideal simplex, which may be identified with the isometry group, and therefore the space of orthonormal frames. But of course this identification is only defined up to conjugacy, which corresponds to where one chooses the base frame to be (Andy has given one possible choice of base frame). The only use that I know of this in higher dimensions is in Gromov’s proof of the Mostow rigidity theorem.

Comment on Mapping classes vs. Homeomorphisms by JL

JL — Wed, 09 Jul 2008 19:54:55 +0000

You’re right, Andy. I just thought, say in dimension 3, the analogous map which assigns each quadruple points of S^2 in general position to a unit tangent vector to H^3. But, of course, the dimensions of the two spaces do not quite match and this map is not injective. So the only reasonable generalization has to be the one you suggested above.

As you remarked, such a generalization does not seem to be useful in the study of hyperbolic structures on manifolds but rather has something to do with Mobius structures on manifolds. See, for example, Kulkarni and Pinkall, Propositions (3.3.ii) and (4.6):

http://www.ams.org/mathscinet-getitem?mr=1273468

Comment on Mapping classes vs. Homeomorphisms by Andy P.

Andy P. — Wed, 09 Jul 2008 15:51:42 +0000

I’m not sure how useful such a parameterization would be in higher dimensions. Let’s just think about 3 dimensions for a minute. Your construction would map a unit tangent vector (x,v) to a pair (a,s), where a is a point on the sphere at infinity and s is a round circle on the sphere at infinity not containing a. The point a would be the endpoint of the geodesic originating at x in the direction of v and s would be the boundary of the geodesic hyperplane through x which is orthogonal to v. This is a bijection (and similarly in higher dimensions).

However, I’m not sure what applications it would have. If M^3 is a finite volume hyperbolic 3-manifold, then one can deduce from Mostow rigidity that the mapping class group of M^3 is exactly the isometry group of M^3. In particular, it is a finite group and it acts on M^3 (no need to pass to the unit tangent bundle). For an infinite volume hyperbolic 3-manifold, one can still lift any diffeomorphism to the universal cover and get an action on S^3. However, that action need not preserve round circles (if it always did, then one could extend Mostow rigidity to infinite volume 3-manifolds, which is wildly false). Thus we would not get an action on our parameterization of the unit tangent bundle.

Comment on Mapping classes vs. Homeomorphisms by JL

JL — Tue, 08 Jul 2008 22:30:14 +0000

“… and let v be the unit tangent vector based at p going in the direction of G.”

Wouldn’t it better if v is “in the direction of K” so that this construction generalizes to higher dimensions?

Comment on Mapping classes vs. Homeomorphisms by Andy P.

Andy P. — Tue, 08 Jul 2008 01:15:22 +0000

The construction goes as follows (I haven’t yet figured out how to use TeX in my comments…sorry). By the space of ideal triangles, we really mean the space of ideal triangles with a counterclockwise ordering on the vertices. This may be identified with the space T of triples (x,y,z) of distinct points on the unit circle so that x–>y–>z goes around the circle in the counterclockwise direction. The map f:T–>UH is the following one. Consider (x,y,z) in T. Let G be the oriented geodesic from y to z. There is then a unique geodesic K originating from x which intersects G at a right angle. Let p be the point of intersection of K and G and let v be the unit tangent vector based at p going in the direction of G. We then define f(x,y,z)=(p,v).

Comment on Mapping classes vs. Homeomorphisms by Jesse Johnson

Jesse Johnson — Mon, 07 Jul 2008 18:14:11 +0000

How does an ideal triangle give you a point in the unit tangent bundle?