# Injectivity of the simple closed curves under geometric intersection number

Let $$\Sigma$$ be a closed surface of genus $$g\geq 2$$ and $$\mathcal{C}$$ be the set of all free homotopy classes of simple closed curves in $$\Sigma$$. Define $$i:\mathcal{C}\rightarrow \mathbb{R}^{\mathcal{C}}$$ by $$i(x)(y)=i(x,y)$$ for $$x,y\in \mathcal{C}$$ where $$i(x,y)$$ is the geometric intersection number.

Q) Does there exist a finite subset $$K\subset\mathcal{C}$$ such that $$i:\mathcal{C}\rightarrow R^{K}$$ is injective?

Yes, there is a collection of $$9g - 9$$ curves that suffices. See Expose Six and Appendix C of "Thurston’s Work on Surfaces" edited by Fathi, Laudenbach, and Poenaru.

Here is a very short sketch - the details are complicated. Suppose that $$\Sigma$$ is the given surface of genus $$g$$. Let $$P = \{\gamma_i\}$$ be a pants decomposition of $$\Sigma$$. We first recall the Fenchel–Nielsen coordinates. Suppose that $$\sigma$$ is a hyperbolic metric on $$\Sigma$$. For every $$\gamma_i \in P$$ we then have

• a hyperbolic length $$\rm{len}(\gamma_i, \sigma)$$ and
• a twist parameter $$\rm{tw}(\gamma_i, \sigma)$$.

The former is a positive real number and the second is a signed real number. Care must be taken to understand the twist as a signed real number and not just as a circle valued number. We can recover the twist from length data as follows. For each $$i$$, pick dual curves $$\alpha_i$$ and $$\beta_i$$ so that the triple of curves $$\{\alpha_i, \beta_i, \gamma_i\}$$ meet each other non-trivially and minimally, and meet the rest of $$P$$ minimally. Then the lengths of these $$9g - 9$$ curves recover the twist parameters.

We now can discuss the original question. Again we have $$\Sigma$$ and $$P$$. We now recall the Dehn-Thurston coordinates. Suppose that $$\delta$$ is a simple closed curve. For every $$\gamma_i \in P$$ we then have

• an intersection number $$\rm{i}(\gamma_i, \delta)$$ and
• a twist parameter $$\rm{tw}(\gamma_i, \delta)$$.

The former is a non-negative integer and the second is an integer. Care must be taken to understand the twist. It can be defined using from intersection data with dual curves, chosen as above.

• I am considering the map from simple closed curves. $9g-9$ theorem is valid for the map from Teichmuller space. Am I missing something? – tessellation Apr 6 '19 at 16:11
• @tessellation Yes it applies. You might also be interested in Bonohan's paper on geodesic currents – Paul Plummer Apr 7 '19 at 17:05
• @PaulPlummer Thanks for the confirmation. I was actually reading Bonohan's paper. I guess I am missing something. – tessellation Apr 7 '19 at 17:21