Given a closed surface of genus $g\geq 2$ and a fixed hyperbolic metric on it, how many pants decompositions exist for that surface?

I tend to believe that it is finite ? For example, if we take a surface of genus 2, and fix a hyperbolic metric on it, then aren't there exactly two ways of cutting it into two pants? I know one can give ( Dehn ) twists along a geodesic which we cut, but would that not change the metric?


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    $\begingroup$ The surface of genus two has only two pants decompositions up to homeomorphism. It has infinitely many pants decompositions up to isotopy. It is important to understand the difference between these two concepts. Check out the Wikipedia pages! The hyperbolic metric doesn't effect the above two statements. $\endgroup$ – Sam Nead Sep 27 '10 at 20:40
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    $\begingroup$ Perhaps the other fact to mention is that, given any topological pants decomposition (defined up to isotopy), you can replace the cutting curves by geodesic representatives. In this sense, you can make any decomposition you like compatible with the metric. $\endgroup$ – HJRW Sep 27 '10 at 21:29
  • $\begingroup$ which wikipidea page are you talking about ? I searched it but apparently I didnt find any detailed treatment of it. $\endgroup$ – Analysis Now Sep 28 '10 at 17:30
  • $\begingroup$ en.wikipedia.org/wiki/Homotopy is the first hit on google for the search "isotopy". mathworld.wolfram.com/Isotopy.html is the third hit. Hmmm. Perhaps you would be better off reading the material in Rolfsen's book on curves in the two-torus (which has pictures and exercises). $\endgroup$ – Sam Nead Sep 28 '10 at 20:33

If the surface is not a pair of pants, there are infinitely many different isotopy classes of pants decompositions (which, as mentioned in the comments, can be made compatible with a constant curvature metric).

They are all "connected" by a finite sequence of "moves". The collection of the isotopy classes (thought of as 0-cells), connected by edges (1-cells) if they are related by the aforementioned moves, and with 2-cells added according to relations among the moves forms a 2-complex that Hatcher has shown to be simply connected. See here for precise definitions and the proof.

On the other hand, the mapping class group acts on the 1-skeleton of this complex (see here), and the quotient is a finite graph. So there are finitely many homeomorphism classes of pants decompositions.

As far as I can tell, the precise number of homeomorphism classes of pants decompositions is not presently known. See here for a lower bound however.

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    $\begingroup$ The paper of Hatcher referred to was published (in slightly modified form) as Section 2 of a paper by Pierre Lochak, Leila Schneps, and myself: "On the Teichmüller tower of mapping class groups", J. reine angew. Math. 521 (2000), 1-24. The proof of the result mentioned in Sean Lawton's answer was based heavily on parts of an earlier paper of Bill Thurston and myself. $\endgroup$ – Allen Hatcher May 17 '16 at 15:31

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