nLab Fenchel-Nielsen coordinates



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 ξ=1\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 gg with nn marked points will have

  • 2g2+n2g - 2 + n generalized pants;

  • 3g3+n3 g - 3 + n closed curves.

The boundary lengths i +\ell_i \in \mathbb{R}_+ and twists t it_i \in \mathbb{R} of these pieces for

1i3g3+n 1 \leq i \leq 3g-3+n

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

Also use θ i:=t i/ i/\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}

This constitutes is a real analytic atlas of Teichmüller space. On MM this reduces to coordinates t i/ it_i \in \mathbb{R}/{\ell_i \mathbb{Z}}, and these constitute a real analytic atlas of moduli space.


  • 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.

Last revised on September 9, 2010 at 19:28:29. See the history of this page for a list of all contributions to it.