2-28【杨文】Oriented graphs of curve complex

If a hyperbolic surface is compared to a piece of music, its curve complex is like a blank sheet music. Each vertex in the curve complex represents a position where a musical note can be placed, and two vertices connected by an edge correspond to adjacent positions on the musical score. By filling these positions with the hyperbolic lengths of curves as notes, the original hyperbolic surface can be uniquely reconstructed from the resulting score. We show that the actual notes are not needed-knowing only whether each note is higher or lower than each of its neighbors is enough to recover the score. Our theorem reveals a new aspect of the rigidity of hyperbolic surfaces, distinct from the 9g-9 theorem and the length spectrum rigidity. This is joint work with Dong Tan.