We investigate a type of distance between triangulations on finite-type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism, and our main results are upper bounds on the distance between triangulations that only depend on the topology of the surface.
"Simultaneous Flips on Triangulated Surfaces." Michigan Math. J. 67 (3) 451 - 464, August 2018. https://doi.org/10.1307/mmj/1530151253