3
votes
1answer
134 views

Change of coordinates for Teichmüller space of the 4-holed sphere

The diagram below indicates 2 ways to use Fenchel-Nielsen coordinates to parameterize the Teichm├╝ller space of conformal structures on the 4-holed sphere with totally-geodesic boundary, corresponding ...
4
votes
1answer
175 views

What's the height of the capped hyperbolic pants?

Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_1,l_2,l_3$. Cap off boundary 1 with a conformal disk. The result is a conformal cylinder, which has a unique flat ...
16
votes
2answers
2k views

A question about the proof of Mostow rigidity

I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma ...
6
votes
1answer
463 views

Analogy of Liouville conformal mapping theorem with Mostow rigidity?

I often hear mention of two theorems, Mostow's rigidity theorem and Liouville's theorem on conformal mappings, which superficially sound similar: they say that a set of geometric structures is, if ...