Modular flip-graphs of one-holed surfaces

We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus g with a single boundary curve and n marked points on this curve and consider triangulations up to homeomorphism with the marked points as their vertices. Our results are bounds on the maximal distance between two triangulations. Our lower bounds assert that these distances grow at least like 5n∕2 for all g≥1. Our upper bounds grow at most like [4−1∕(4g)]n for g≥2, and at most like 23n∕8 for the bordered torus.