Proof that the length function $\ell: \operatorname{Teich}(S) \to \mathbb{R}^\mathcal{S}$ is injective without the $9g-9$ theorem

In Chapter 10 about Teichmüller spaces of Farb and Margalit's "A Primer to Mapping Class Groups", the length function $$\ell: \operatorname{Teich}(S) \to \mathbb{R}^\mathcal{S}$$ is described, where $S$ is a compact surface with finitely many punctures and $\chi(S) < 0$ and $\mathcal{S}$ denotes the set of isotopy classes of essential simple closed curves in $S$.

It is already impressive that the map $\ell$ is injective (which they also remark), however they prove the much stronger $9g-9$ theorem.

Question: Is there a (simpler) proof that $\ell$ is injective without showing that the length function is actually determined by its value on only $9g-9$ simple closed curves?

I've searched for a bit now, and came across e.g. these lecture notes, where Theorem 3.3 claims that the map is injective. The reference given is "Travaux de Thurston sur les surfaces", in which Theorem 7.9 states that $\ell$ is injective. However, it is not proven independently from the $9g-9$ theorem after all.

• Your last sentence is worded strangely: Theorem 7.9 is proven, because it is a corollary of the $9g-9$ theorem. Jul 29, 2016 at 14:34
• @LeeMosher: Yes, I realize that it is a corollary, what I mean is that it is not explicitly proven (even though stated) independently from the $9g-9$ theorem.
– Huy
Jul 29, 2016 at 14:35

1 Answer

There is a representation-theoretic viewpoint, where one can consider traces of $SL_2(\mathbb{R})$ representations (determined by lengths of geodesic up to sign). Then one sees that the trace variety is determined by traces of simple closed curves, modulo certain relations. See papers of Feng Luo, for example. He shows that the modulus of the surface is determined by the moduli of all of the 1-punctured tori and 4-punctured sphere subsurfaces. In turn, it is not hard to show that the modulus of a once punctured torus is determined by the lengths of 3 curves intersecting pairwise once (the complementary regions are two triangles with vertices on the 3 Weierstrass points, and an annulus: the triangle has sidelengths half the geodesic lengths, and then one can see that the lengths determines the geometry of the triangles and annulus; there is a similar picture for 4-punctured spheres).

See this question for more references on parameterizing Teichmuller space by geodesic lengths.