7
$\begingroup$

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose angles are all right angles such that in their natural circular order the sides are $(a,x,b,y,c,z)$ for some positive real numbers $x,y,z$.

I know two sources that claim to prove this. In Hubbard's book "Teichmuller theory, volume 1" he has a proof, but it rests on Exercise 3.5.5 (which he describes as "surprisingly tricky", and which I can only solve with a really terrible and unenlightening calculation). In Farb-Margalit's "Primer on mapping class groups", there is also a proof, but they essentially assert something equivalent to that exercise without proof.

Question: Does anyone know a source with a complete proof? Even better, a proof that minimizes terrible calculations.

$\endgroup$
7
  • $\begingroup$ "Ratcliffe, Foundations of Hyperbolic Manifolds is decent. (See the section on hyperbolic trigonometry.) " quote from comment at math.stackexchange.com/questions/1076278/… $\endgroup$
    – Will Jagy
    Nov 1, 2021 at 1:33
  • $\begingroup$ @WillJagy: I had forgotten about that one. Looking at it, it does give all the details, but via a gigantic wall of formulas (sadly, par for the course in that book). Hopefully someone will know a more enlightening way of doing it... $\endgroup$
    – Lisa
    Nov 1, 2021 at 1:38
  • $\begingroup$ Apparently Thurston gave a very simple "Law of Sines" for right angled hexagons, see mathoverflow.net/questions/278365/… I will check, my guess is Thurston gave a proof that I would, well, like. $\endgroup$
    – Will Jagy
    Nov 1, 2021 at 2:45
  • $\begingroup$ library.msri.org/books/gt3m/PDF/2.pdf $\endgroup$
    – Will Jagy
    Nov 1, 2021 at 2:49
  • $\begingroup$ Alright, Thurston concludes with a couple of diagrams after presenting the hyperbolic Laws of sines, and two cosines. Those are in my little article zakuski.utsa.edu/~jagy/papers/Intelligencer_1995.pdf $\endgroup$
    – Will Jagy
    Nov 1, 2021 at 2:58

3 Answers 3

4
$\begingroup$

My personal favorite proof is described well in this blog post (which attributes it to Hermann Karcher, though I first heard a version of it back in graduate school, so it should probably just be called folklore). It is entirely synthetic and calculation-free.

$\endgroup$
2
$\begingroup$

Section 7.19 (Hexagons) of Beardon's book The geometry of discrete groups gives a proof.

$\endgroup$
3
  • $\begingroup$ Sam, does it show Thurston's claim about right angled pentagons, if $A,B$ are lengths of two consecutive sides, and $D$ the side that has no common vertex with either of those two, then $\cosh D = \sinh A \sinh B \; ? \; $ This is 2.6.17 on the final page of library.msri.org/books/gt3m/PDF/2.pdf , where 2.6.18 is his Law of Sin(h)es for right angled hexagons. It ought to come from clever drawings and a little trig, I don't see it. $\endgroup$
    – Will Jagy
    Nov 1, 2021 at 17:53
  • $\begingroup$ Dear Will - Yes. It is in Section 7.18 (Pentagons). $\endgroup$
    – Sam Nead
    Nov 2, 2021 at 16:40
  • $\begingroup$ Thank you. I have books about non-Euclidean plane geometry owing to contact with Hartshorne and Greenberg. Neither book appears to do any pentagons and hexagons, but both Lambert and Saccheri quadrilaterals. Oh, I found Beardon and printed out 7.17-7.19 $\endgroup$
    – Will Jagy
    Nov 2, 2021 at 17:25
0
$\begingroup$

A proof of this that I enjoyed reading appears as lemma 3.6 in Thurston's Work on Surfaces, by Fathi, Laudenbach, and Poénaru.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.