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; we consider triangulations up to homeomorphism with the marked points as their vertices. Our main results are upper and lower bounds on the maximal distance between triangulations depending on $n$ and can be thought of as bounds on the diameter of flip-graphs up to the quotient of underlying homeomorphism groups. The main results assert that the diameter of these quotient graphs grows at least like $5n/2$ for all $g\geq 1$. Our upper bounds grow at most like $[4 -1/(4g)]n$ for $g\geq 2$, and at most like $23n/8 $ for the torus.