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I am a first year student and a learner of hyperbolic geometry. I was wondering if you could suggest some exciting topics to research about in this field (some people suggested fundamental polygons and areas of hyperbolic triangles).

Any other exciting topics to be suggested? I am a first year student, but I don't mind having to slog through some groups and real/complex variables.

What I'm looking for, is I have said above is a topic that I can research on. E.g. I could try to do something like the analogue of the euler lagrange equations in the euclidean plane, namely instead of minimising the functional defined by $\int_a^b \sqrt{1+ \Big(\frac{dy}{dx}\Big)^2} dx$, I could try and minimise the functional that defines the hyperbolic metric in $\mathbb{H}$ and see what kind of equations I get out of that.

Ben

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Do you want to restrict to 2-dimensional hyperbolic geometry or are you studying arbitrary-dimensional hyperbolic geometry? –  Ryan Budney Apr 8 '11 at 3:03
    
2-dimensional, preferably. –  user38268 Apr 8 '11 at 6:23
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Seeing as you do not mind learning some complex analysis, I would recommend the book "Visual complex analysis" by Tristan Needham. This book has a chapter on hyperbolic geometry that seems to be at an appropriate level.

Edit: Google Books has a preview where you can see the table of contents and some of the material here.

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Hi thanks, some people did suggest to me this book before. However, I was wondering if some people could suggest some topics (someone more experienced than I) to research on; certainly not too advanced till it deals with manifolds and stuff! –  user38268 Apr 8 '11 at 14:05
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Try reading "Glimpses of Algebra and Geometry" by Toth - the later chapters have material that may be relevant, and it's a relatively friendly introduction.

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If you want to get a complete undergraduate text in which you can find a metric approach, I suggest you to find and see this book:

Geometry: A Metric Approach with Models by Richard S. Millman.

Last semester, I taught some chapters to students and believe me they were being exited of working with this geometry . Of course, I used some software simultaneously like this one NonEuclid.

enter image description here

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