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Geodesics in $\mathbb H^2$ have the following properties: For every two points in the plane there exists a unique geodesic joining them. Every geodesic determines exactly two points on the boundar …

Let $S$ be an orientable surface with negative Euler characteristic. Can somebody provide a reference for the following well-known results: the cyclic subgroup generated by a pseudo-Anosov element …

Let $S$ be an orientable surface of genus $g$ with $b>0$ boundary components, and let $\mathrm{Mod}(S)$ be its mapping class group, that is, the group of isotopy classes of its homeomorphisms modulo i …

Let $S_{g,b}$ be an oriented surface with $b$ boundary components and $S_g^b$ be an oriented surface with $b$ punctures. Let $\mathrm{Mod}(S_{g,b})$ and $\mathrm{Mod}(S_g^b)$ their (orientation preser …

A very nice paper about the Teichm\"uller space of non-orientable surfaces, Fenchel-Nielsen coordinates and other generalizations of Thurston's theory to non-orientable surfaces is this one by Papadop …

What is the centralizer of a pseudo-Anosov element in the mapping class group of an orientable punctured surface? Is it cyclic? If so, where can I find a proof?

Does there exist a quasi-isometric embedding $$MCG(S) \to (\mathrm{Teich}(S), d)$$ for $d$ any "known" distance on the Teichmuller space (i.e. Teichmuller, Weil-Petersson, Thurston...) ?

Let $MCG(S_{g,b}^s)$ be the mapping class group of a surface $S_{g,b}^s$. Assume that it is not trivial. Is it true that $MCG(S_{g,b}^s)$ is isomorphic to $MCG(S_{g',b'}^{s'})$ if and only if $ S_{g, …

Q1. A beautiful survey about these results is the following: McCarthy-Papadopoulos, Simplicial actions of mapping class groups, in Handbook of Teichmüller theory Volume III Q2. Luo's proof of t …