# Number of curves in an admissible system of Jordan curves on a surface

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?

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 $$(g+1)+2(g-2)=3g-3.$$