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;

• $3 g - 3 + n$ closed curves.

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

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

constitute the Fenchel-Nielsen coordinates on Teichmüller space $\Tau$.

Also use $\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}$

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