9g-9 theorem

In mathematics , the theorem is a proposition from Teichmüller's theory . ${\displaystyle 9g-9}$

It says that there are a number of closed curves on a ( closed , orientable ) surface of the gender , the lengths of which clearly define every hyperbolic metric on the surface. ${\displaystyle g}$ ${\displaystyle 9g-9}$

The theorem provides an embedding of the Teichmüller space in the , which, however, is not surjective . Instead, a diffeomorphism of the Teichmüller space is realized by the Fennel-Nielsen coordinates , which assign the length and the twist parameters of the corresponding geodesics to a selected set of closed curves. ${\displaystyle \mathbb {R} ^{9g-9}}$ ${\displaystyle \mathbb {R} ^{3g-3}\times \mathbb {R} _{+}^{3g-3}}$ ${\displaystyle 3g-3}$

The theorem is generalized to (orientable) areas of gender with and points , where the lengths of closed geodesics are required. ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle 3(3g-3+n)}$

Hamenstädt has shown that in the case of closed areas, even the lengths of closed geodesics can determine the hyperbolic metric, while the lengths of geodesics are not sufficient. Geodesics are required for areas with peaks . ${\displaystyle 6g-5}$ ${\displaystyle 6g-6}$ ${\displaystyle 6g-5+2n}$

literature

• Benson Farb, Dan Margalit: A primer on mapping class groups. (= Princeton Mathematical Series. 49). Princeton University Press, Princeton, NJ 2012, ISBN 978-0-691-14794-9 . ( online ; pdf)
• Ursula Hamenstädt: Length functions and parametrization of Teichmüller space for surfaces with cusps. Ann. Acad. Sci. Fenn. Math. 28: 75-88 (2003).