According to Mathworld, hyperbolic geometry is well understood in 2 dimensions but not in 3 dimensions.


What isn't well understood about hyperbolic geometry in 3D, that is in 2D or that is in 3D-Euclidean geometry?

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    $\begingroup$ I wouldn't take that comment on Wolfram's site too seriously..... $\endgroup$ – Robin Chapman Oct 26 '10 at 9:49

I don't know what "well-understood" means in this context. It is true that hyperbolic geometry is a very rich area in both dimensions two and three.

In dimension two every closed surface admitting one hyperbolic structure admits uncountably many, and these fit together to form moduli space. Moduli space is an object of intense study from the point of view of algebraic geometry, number theory, dynamical systems, etc. The "fundamental group" of moduli space is the mapping class group (aka the modular group) and this is again studied by many different groups.

In dimension three we have Mostow rigidity saying that a closed three-manifold admits either zero or essentially one hyperbolic structure. It follows that the geometric invariants of the hyperbolic structure give topological invariants of the three-manifold, which was a wonderful shock to topologists in the 70's. (I was recently told a story about Thurston wandering around a conference at Warwick, in the 70's, with a programmable calculator on his belt, saying that he was computing volumes of knots in the three-sphere. Nobody had any idea what he was saying, apparently.)

It follows that the moduli spaces of hyperbolic three-manifolds are much more simple. To get an interesting theory one allows hyperbolic structures that are not complete (or discrete, etc) and then the moduli spaces become very wild and are not well understood.

  • $\begingroup$ I think the comment is more along the lines of "H^2 is well-understood but H^3 is not." Whatever that means. $\endgroup$ – Qiaochu Yuan Oct 26 '10 at 12:32
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    $\begingroup$ Well, it probably doesn't mean anything. So I took it as an opportunity to spread the joy. $\endgroup$ – Sam Nead Oct 26 '10 at 16:49

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