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I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here:


I am trying to understand the article by Maryam Mirzakhani about Simple geodesics and Weil-Petersson volumes. In the third section of this article, the following proposition is stated. And I want to know why this proposition is correct. I mention the proposition:

Proposition: Let $P$ be a hyperbolic pair of pants with (non-empty)geodesic boundary components $\beta_1 , \beta_2, \beta_3$ of lenghts $x_1, x_2, x_3$ respectively. Then $P$ contains $5$ complete geodesics disjoint from $\beta_2 , \beta_3$ and orthogonal to $\beta_1$. More precisley, two of these geodesics meet $\beta_1$ respectively at $y_1,y_2$ and spiral to $\beta_3$, the other two meet $\beta_1$ respectively at $z_1,z_2$ and spiral to $\beta_2$. there is also a unique common geodesic perpendicular from $\beta_1$ to itself meeting $\beta_1$ perpendicularly at two points $w_1, w_2$.

enter image description here

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I suggest that you look at what happens in the universal cover given by the Poincare disk model. The five geodesics are quite easy to see. In the picture below, the greyed zone is a fundamental domain for the pair of pants. The five sought geodesics are in bold. The book of Peter Buser, geometry and spectra of compact Riemann surfaces, is a good reference concerning the representation of surfaces as quotients of the Poincare disk.

five-geodesics-in-poincare-disk

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  • $\begingroup$ Very nice explanation, but the picture is a bit sloppy. If someone unfamiliar with hyperbolic geometry sees it, they might get wrong ideas. For example, two geodesics perpendicular to the geodesic $\beta_1$ cannot have a common endpoint at infinity unless they are equal. The shaded area seems to be a fundamental domain for the pair of pants with ends attached to them (like a "pair of flares", to stay in jargon). $\endgroup$ – Sebastian Goette Sep 16 '20 at 19:49
  • $\begingroup$ Yes indeed this is a quick sketch. I edited the picture a bit. The fundamental domain includes the three funnels as you said. $\endgroup$ – coudy Sep 16 '20 at 20:06
  • $\begingroup$ @coudy Thank you, but I think you mean the book "Geometry and Spectra of Compact Riemann Surfaces". Am I correct? $\endgroup$ – Amirhossein Sep 19 '20 at 16:46
  • $\begingroup$ Yes, indeed. Corrected. $\endgroup$ – coudy Sep 19 '20 at 20:50

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