As far as I know, the most common structure of curves in surface is called Curve the curve complex. John Hempel linked the curve complex and Heegaard Splitting and defined Heeggaard Heegaard Distance. There are lots of results concering on it. You can see about that, e.g., the work of Tseuyoshi Kobayashi, Ruifeng Qiu, Martin Scharlemann, Saul Schleimer, Maggy tomovaTomova, Yair Minsky and so on.But the weak part of these
This structure is has a weak point in that that you can not see any symmetry, and since it is not locally finite, we can not figure out the geodesic. My question is:
Is there any other structure which can avoid the weak part? Or, can you say your favorite structurepoints?Thank you.
Edit: I have changed it. Thanks for advice.