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

- they are nonintersecting with each other;
- no two are freely homotopic;
- 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?