Rotation Distance, Triangulations of Planar Surfaces and Hyperbolic Geometry

In a beautiful paper, Sleator, Tarjan and Thurston solved the problem of maximum rotation distance of two binary trees. Equivalently they solved the problem of rotation distance of triangulations on the disk. We extend their results to rotation distance of triangulations of other planar surfaces. We give upper and lower bounds for this problem. Equivalently, by duality, one can interpret our results as bounds for rotation distance of the dual graphs of the triangulation graphs. They are the counter parts to binary trees in the case of disk. In the case of the annulus, by cutting along an edge between the inner and outer boundary circles, we obtain rooted binary trees with a distinguished path to a leaf. The upper bound is obtained by looking at the triangulations in the universal covering space, and the lower bound is obtained by extending and applying the technique of volume estimate in hyperbolic geometry.