On diffeomorphisms over surfaces trivially embedded in the 4-sphere.
We study the ideal triangulation graph of an oriented punctured surface of finite type. We show that if is not the sphere with at most three punctures or the torus with one puncture, then the natural map from the extended mapping class group of into the simplicial automorphism group of is an isomorphism. We also show that the graph of such a surface , equipped with its natural simplicial metric is not Gromov hyperbolic. We also show that if the triangulation graph of two oriented punctured...