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

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

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

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

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

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