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Consider a compact Riemann surface of genus $g\geq2$. An admissible system of Jordan curves is a finite collection of Jordan curves $\{\gamma_1,\cdots,\gamma_n\}$ such that

  1. they are nonintersecting with each other;
  2. no two are freely homotopic;
  3. none is homotopic to 0

This concept is mentioned in Masur's on a class of geodesic in teichmuller space and in this paper the author mentions that an admissible system has at most $3g-3$ Jordan curves. I am wondering why and how to prove it.

My attempt would be to consider a fundamental polygon with $4g$ sides, consecutively labeled by $$a_1,b_2,a_1^{-1},b_1^{-1},\cdots,b_g^{-1}$$ Now choose a point $p$ on the edge $a_1$, then $p$ appears on $a_1^{-1}$ as well. Connecting them gives a closed curve. We cannot do the same for $b_1$, but we can do this for $a_2,a_3,\cdots,a_g$. This gives $g$ curves. Finally an edge is also a closed curve. Now any curve not null-homotopic should cross an edge of the fundamental polygon, but I cannot seem to find another one without intersecting the existing ones. So I can find at most $g+1$ curves. Where do the $3g-3$ curves come from?

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It comes from the so called "pants decomposition". A "pair of pants" (or simply pants) is a sphere with $3$ holes. Every compact surface of genus $g\geq 2$ can be decomposed into such pants. The $3g-3$ curves are the pants boundaries.

To visualize, represent your surface in the following way. Consider the sphere with $g+1$ holes, and your surface is obtained by gluing this to its double (mirror image). Now decompose the sphere with $g+1$ holes into pants. To do this you draw $g-2$ disjoint closed curves. The mirror image will contain the same number of them. So the total is $$(g+1)+2(g-2)=3g-3.$$

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