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 . These building blocks (of hyperbolic 2d geometry) are precisely
the 3-holed sphere;
the 2-holed cusp;
the 1-holed 2-cusp;
Each surface of genus with marked points will have
The boundary lengths and twists of these pieces for
1 \leq i \leq 3g-3+n
constitute the Fenchel-Nielsen coordinates on Teichmüller space .
This constitutes is a real analytic atlas of Teichmüller space. On this reduces to coordinates , 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.
Revised on September 9, 2010 19:28:29
by Zoran Škoda