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Suppose $S$ is a surface and the Mapping Class Group, $Mod(S)$, of $S$, is the group of self-homeomorphisms of $S$, up to isotopy. This group acts on a graph called "curve graph", denote by $\mathcal{C}(S)$. The construction of this graph can be found in "A primer on mapping class groups" by Benson Farb and Dan Margalit.

I was studying the paper by Masur and Minsky: Geometry of the complex of curves I: Hyperbolicity, where they proved that curve graph is hyperbolic in the sense of Gromov. In the introduction of the paper they have written and I quote: "it is plain that $Mod(S)$ acts isometrically on $\mathcal{C}(S)$, with compact quotient."

I was finding difficulty to understand that why there will be compact quotient, and what is the fundamental domain.

Thanks in advance.

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    $\begingroup$ This follows from the “change of coordinates principle” described in Farb-Margalit’s primer. $\endgroup$ Commented Apr 9, 2024 at 11:04
  • $\begingroup$ the change of coordinate principle says that between two distinct, non-separating curves there is a homeomorphism sending one to other. does that mean there is only one orbit? $\endgroup$ Commented Apr 9, 2024 at 11:19
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    $\begingroup$ That’s just one example of the change of coordinates principle. Go back and read that chapter of the primer. $\endgroup$ Commented Apr 9, 2024 at 11:54
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    $\begingroup$ (a good test of your understanding of it is that the cocompactness of the action becomes obvious, though it is a bit annoying to write down a fundamental domain) $\endgroup$ Commented Apr 9, 2024 at 11:55
  • $\begingroup$ A meaningful question would be about cardinality of the simplex set of the curve complex modulo the mapping class group (or, better, the pants complex), generalizing the Catalan numbers. I do not know an answer to this one but it might be in the literature. $\endgroup$ Commented Apr 9, 2024 at 15:58

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As Andy says, that the action is co-compact follows from the "change of coordinates principle". Finding a fundamental domain for $S_2$ (the compact, connected, oriented, boundary-less surface of genus two) is a nice exercise. The problem of finding a fundamental domain for $S_g$ quickly succumbs to "combinatorial explosion".

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