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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?

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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.

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

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