Abstract
The ne curve graph of a surface was introduced by Bowden, Hensel, and Webb as a graph consisting of the actual essential simple closed curves on the surface. Long, Margalit, Pham, Verberne, and Yao proved that the automorphism group of the ne curve graph of a closed orientable surface is isomorphic to the homeomorphism group of the surface. We generalized their result to closed nonorientable surfaces Ng of genus g 4. This is a joint work with Mitsuaki Kimura.
Background
We define the fine curve graph. Feiner Kurvengraph. Then we define the curve graph and give examples as well as nonexamples.
Homeomorphisms act on the curve graph and the fine curve graph.
Theorem
We state the main theorem von Yao. (Theorem for orientable surfaces) We define the extended fine curve graph. Here is easier to show the isomorphism between the automorphism and the the homeomorphismus. We define a composition and show that this is actually just the identity.
Issues for nonorientable surfaces
A two sided curve is a curve whose neighbourhood is homoemorphic to an annulus. We define a hull of a set of curves.
There is a Lemma by Long-Margalit-Pham-Verberne-Yao which is important to show the isomorphism.
Sadly the proof breaks down in nonorientable surfaces. To fix this a condition is added. This makes the proof work again.