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The answer is yes. It's a little easier if we consider oriented geodesics (which is fine for your argument). Let $\alpha$ and $\beta$ be two oriented simple closed curves on $S_g$ that intersect onc …

The original proof of Helly's theorem was topological and only uses basic homological properties of convex sets. It generalizes to all sorts of contexts, including the one you are interested in. Her …

There is a huge literature, and I'm not sure exactly what you are looking for. That being said, Masur-Tabachnikov's survey "Rational billiards and flat structures" and Masur's survey "Ergodic Theory …

This isn't true. Here's one easy construction. For some $g \geq 2$, let $\Sigma_g$ be a closed oriented genus $g$ surface and let $f\colon \Sigma_g \rightarrow \Sigma_g$ be a homeomorphism represent …

My personal favorite proof is described well in this blog post (which attributes it to Hermann Karcher, though I first heard a version of it back in graduate school, so it should probably just be call …

If $M$ is a closed $3$-manifold and $\pi_1(M) \cong A \ast B$ with $A$ and $B$ nontrivial, then Kneser's conjecture (which is a theorem -- the proof can be found in Hempel's book on 3-manifolds) says …

Chapter 2 of Lando and Zvonkin's lovely book "Graphs on Surfaces and Their Applications" contains a nice, down-to-earth exposition of the proof of Belyi's theorem together with a large number of expli …

They have all been resolved in some fashion with the exception of problem 23, which asserts that the volumes of all hyperbolic 3-manifolds are not rationally related. We know basically nothing about …