# How many pants decompositions for a given surface with a fixed hyperbolic metric?

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?

Thanks!

• 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. – Sam Nead Sep 27 '10 at 20:40
• 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. – HJRW Sep 27 '10 at 21:29
• which wikipidea page are you talking about ? I searched it but apparently I didnt find any detailed treatment of it. – Analysis Now Sep 28 '10 at 17:30
• 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). – Sam Nead Sep 28 '10 at 20:33