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I'm a first year Graduate student.In my first semester complex analysis course there was a topic hyperbolic geometry but due to time limit unfortunately this topic was not touched during the course.Now i am trying to read this topic on my own from the point of view of complex analysis.I tried to find out some books but could not find much,the only book which i found was Gamelin's complex analysis.This book also does not contain,it discuss some hyperbolic geometry on disc only.

What are the interesting result in complex analysis related with hyperbolic geometry? Are there any good notes/book which contains the same material?

Thank you

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    $\begingroup$ Does the down voter care to explain? $\endgroup$
    – Math Lover
    Commented Dec 18, 2016 at 10:11

4 Answers 4

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Krantz Complex Analysis: The Geometric Viewpoint has some material on this. You can find e.g a geometric version of the Schwarz lemma.

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More a long comment

The problem is a bit that only a small part of complex analysis is applicable to hyperbolic geometry. There is an overlap between the subjects but both are larger than the intersection, Maybe a good starting point is the study of mobius transformations but then again most of it has no relationship to hyperbolic geometry and it requires quite a lot of study to select what is relevant and what is not.

but still to end with some references:

1 ) Stahl, the Poincaré's half plane (lot on hyperbolic geometry, not a lot on mobius transformations)

2) Swerdtfeger, geometry of complex numbers (lot o mobius transformations, not a lot on hyperbolic geometry)

But at some points even these two disagree with each other

added later

maybe also

3) Alan Beardon, The geometry of discrete groups

(goes into higher dimensional mobius transformations, not a lot on hyperbolic geometry)

Sadly i am not awary of a good book that has none of the deficiencies)

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  • $\begingroup$ Thanks a lot,i'll have a look! $\endgroup$
    – Math Lover
    Commented Dec 27, 2016 at 11:11
  • $\begingroup$ Hopefully there are better books than the two I mentioned let us know how you get along $\endgroup$
    – Willemien
    Commented Dec 27, 2016 at 17:31
  • $\begingroup$ Dear @Willemien: I have started reading the mentioned book,please feel free to update your answer if you have any better book in your mind.Thank you! $\endgroup$
    – Math Lover
    Commented Dec 28, 2016 at 10:57
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Plane hyperbolic geometry can be represented in the complex half-plane(Poincare half-plane model). This produces certain connexions between complex functions, complex domains and hyperbolic geometry.

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The hyperbolic geometry of the disk is (in some sense) "all you need", since the Uniformization Theorem shows that every Riemann surface is a holomorphic quotient of either the unit disk, the Riemann sphere, or the complex plane. If the universal cover is the unit disk, then we get a conformally invariant metric on our Riemann surface by pushing down the Poincare metric on the disk.

One place to read this story in some depth is chapter 2 of "the geometry of complex domains" by Green, Kim, and Krantz.

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