# 9g-9-Theorem - 9g-9-Theorem

In math , that is${\displaystyle 9g-9}$-Theorem a theorem from the Teichmüller theory .

It says that it is on a ( closed , orientable ) area of the sex ${\displaystyle g}$ a lot of ${\displaystyle 9g-9}$ closed curves whose lengths uniquely define each hyperbolic metric on the surface.

The sentence thus embeds the Teichmüller space in the${\displaystyle \mathbb {R} ^{9g-9}}$which, however, is not surjective . A diffeomorphism of the Teichmüller space with${\displaystyle \mathbb {R} ^{3g-3}\times \mathbb {R} _{+}^{3g-3}}$is instead realized by the Fennel-Nielsen coordinates , which are a selected set of${\displaystyle 3g-3}$ the length and the twist parameters of the corresponding geodesics are assigned to closed curves.

The sentence is generalized to (orientable) areas by gender ${\displaystyle g}$ With ${\displaystyle n}$and tips where then the lengths of${\displaystyle 3(3g-3+n)}$ closed geodesics are required.

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

## 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).