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Results tagged with hyperbolic-geometry
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user 11142
19
votes
Accepted
Random links and $3$-manifolds
There is no universally accepted model of random three-manifolds (or random knots/links) for that matter, however, hyperbolicity is pervasive in all known models. The most popular (but not really sati …
- 96.4k
19
votes
Why is there a unique hyperbolic simplex of largest area?
Firstly, this is not something you "can easily prove by calculation": The proof (by Haagerup and Munkholm) was published in Acta.
Secondly, in three dimensions, the set of ideal simplices are parame …
- 96.4k
18
votes
17
votes
Accepted
Does Helly's theorem hold in the hyperbolic plane?
I don't understand your reference to the model (since the geometry of the hyperbolic plane does not depend on any model), but, in fact, the Beltrami-Klein model demonstrates that any qualitative state …
- 96.4k
13
votes
How to see isometries of figure 8 knot complement
The figure 8 complement decomposes into 2 regular ideal tetrahedra (see Thurston's notes, here for example.) This gives the involution quite explicitely.
- 96.4k
12
votes
Accepted
Dilogarithm, tetrahedrons, and hyperbolic space
The five term relation comes from the fact that the sum of the volumes of tetrahedra $ABCD$ and $ABCE$ equals the sum of the volumes of the three tetrahedra $ABDE, ACDE, BCDE.$ One can think of $ABCDE …
- 96.4k
12
votes
Tetrahedra with prescribed face angles
The question seems a little confused, in particular since the OP is asking about dihedral angles but is calling them face angles. In any case, despite the gloom in the accepted answer, a lot is known. …
- 96.4k
11
votes
What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their m...
I believe that Riemann realized that the sphere was a model for elliptic geometry (1854), while Beltrami (in 1868) introduced his model for hyperbolic space, but neither Bolyai nor Lobachevsky proved …
- 96.4k
11
votes
Thurston's 24 questions: All settled?
Yes, and yes, and yes. The 1st by Perelman, the 24th by Joseph Maher.
- 96.4k
11
votes
10
votes
Hyperbolic Volume and Chern-Simons
The first reference known to me is
Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.
Howev …
- 96.4k
10
votes
Accepted
9
votes
Arithmetic Fuchsian group
The answer is NO. Any four times punctured sphere admits two involutions, the quotient by which is an orbifold of the signature you describe. Similarly, you can take a punctured torus, and the quotien …
- 96.4k
8
votes
The fundamental group of a closed surface without classification of surfaces?
Yes, the magic words are "The Poincare polygon theorem". For (considerably) more detail, see Fine and Rosenberger "Algebraic generalizations of discrete groups: A path to combinatorial group theory th …
- 96.4k
8
votes
Accepted
Can most 3 dimensional hyperbolic orbifolds with finite volume be covered by a hyperbolic ma...
Yes, this is true for all of them. Any finitely generated matrix group has a torsion-free subgroup of finite index; this is the so-called "Selberg's lemma". A canonical source is Ratcliffe's Hyperboli …
- 96.4k