# Exciting Topics in Hyperbolic Geometry

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. –  Benja Apr 8 '11 at 6:23