nLab
Fenchel-Nielsen coordinates

Idea

The Fenchel-Nielson coordinates are certain coordinates on Teichmüller space.

They parameterize Teichmüller space by cutting surfaces into pieces with geodesic boundaries and Euler characteristic $\xi =-1$. These building blocks (of hyperbolic 2d geometry) are precisely

• the 3-holed sphere;

• the 2-holed cusp;

• the 1-holed 2-cusp;

• the 3-cusp

Each surface of genus $g$ with $n$ marked points will have

• $2g-2+n$ generalized pants;

• $3g-3+n$ closed curves.

The boundary lengths ${\ell }_i\in {ℝ}_{+}$ and twists $t_i\in ℝ$ of these pieces for

$$1\le i\le 3g-3+n$$1 \leq i \leq 3g-3+n

constitute the Fenchel-Nielsen coordinates on Teichmüller space ${\rm T}$.

Also use ${\theta }_i:=t_i/{\ell }_i\in ℝ/\mathbb{Z}$

This constitutes is a real analytic atlas of Teichmüller space. On $M$ this reduces to coordinates $t_i\in ℝ/{\ell }_i\mathbb{Z}$, and these constitute a real analytic atlas of moduli space.

References

• Kathy Paur, The Fenchel-Nielson coordinates of Teichmüller spaces (pdf)
• Werner Fenchel, Jakob Nielsen, reprinted in Discontinuous groups of isometries in the hyperbolic plane, edited by Asmus L. Schmidt; De Gruyter Studies in Math. 29, 2003.