# Pants decomposition and moduli space of $\Sigma_g$ for $g>1$

By Section 8.3.1 of the book: A primer on mapping class groups by Farb and Margalit, a pair of pants is a compact surface of genus 0 with three boundary components. Let $$S$$ be a compact surface with $$\chi(S)<0$$. A pair of pants decomposition of $$S$$, or a pants decomposition of $$S$$, is a collection of disjoint simple closed curves in $$S$$ with the property that when we cut $$S$$ along these curves, we obtain a disjoint union of pairs of pants. Equivalently, a pants decomposition of $$S$$ is a maximal collection of disjoint, essential simple closed curves in $$S$$ with the property that no two of these curves are isotopic. In particular, a pants decomposition of $$\Sigma_g$$ (the orientable closed surface of genus $$g$$) for $$g>1$$ has $$3g-3$$ curves, cutting $$\Sigma_g$$ into $$2g-2$$ pairs of pants.

On the other hand, according to Riemann's computation, the dimension of the moduli space $$\mathcal{M}_g$$ of algebraic curves of genus $$g>1$$ is $$3g-3$$. See, for example, the book: Algebraic curves towards moduli spaces.

My question: why are the number of curves in the pants decomposition of $$\Sigma_g$$ for $$g>1$$ and the dimension of the moduli space of $$\Sigma_g$$ for $$g>1$$ the same number $$3g-3$$? Is there any deep relation?

Thank you!

• Are you looking for a deeper relation than the Teichmüller space? – Aurelio Oct 11 '20 at 9:47
• Shaking hands. Let take real dimension, so the number of curves is $3g-3$, and the dimension is $6g-6$. The moduli essentially consist in choosing the size $\ell_c$ of these $3g-3$ curves $c$, and choose an angle $\alpha_c\in\mathbf{R}/\ell_c\mathbf{Z}$ of pasting along each of these curves, which gives $6g-6$ parameters of freedom (which can all be achieved since one can freely prescribe lengths of boundary curves of pants). A complete proof along these way of thoughts can be found in P. Buser, Geometry of Riemann surfaces. – YCor Oct 11 '20 at 10:07

I am unsure if this is what you are looking for, but the moduli space $$\mathcal{M}_g$$ can be viewed as an orbifold quotient of the Teichmüller space by the mapping class group. I cannot offer more details, but this is explained in a paper by John H. Hubbard and Sarah Koch.

EDIT I notice that this has been addressed in another question as well.

• How does that explain the relation with the number of simple curves? – abx Oct 11 '20 at 10:28
• @abx Indeed I did not give an explanation of the number of simple curves: this is described in the comment above by YCor, which I understand refers to the Fenchel–Nielsen coordinates. I was aiming at giving some insight into why the Teichmüller space and the Riemann moduli space have the same dimension. This is the "deep relation" that I assume the OP was looking for. – Aurelio Oct 11 '20 at 10:40
• this is an essentially trivial relation. I don't think this is the sense of the question. – abx Oct 11 '20 at 11:50
• @abx I agree that the relation is essentially trivial, but the Teichmüller space is not referred to in the original question, so I assumed it was worth mentioning. – Aurelio Oct 11 '20 at 11:54