1
$\begingroup$

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that there is at least one intersection point between the geodesic representative of $\alpha$ and the geodesic representative of $\beta$ which is not a double point.

Q) Is $X$ nowhere dense in $\mathcal{T}(S)$?

I think it is true but I can't prove it.

$\endgroup$
1
  • 1
    $\begingroup$ One has to rule out 3 simple closed curves on a punctured torus meeting at a point, since these will always have a triple intersection at the fixed point of an elliptic involution which sends each curve to its inverse. $\endgroup$
    – Ian Agol
    Oct 15, 2015 at 21:25

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.