Modular flip-graphs of one holed surfaces

TL;DR

This paper investigates the structure and size of flip-graphs of triangulations on one-holed surfaces, providing bounds on their diameters that depend on the genus and number of marked points, with implications for understanding their geometric complexity.

Contribution

It establishes new upper and lower bounds on the diameter of flip-graphs for one-holed surfaces, extending understanding of their geometric properties and complexity.

Findings

Diameter grows at least like 5n/2 for all g≥1.

Upper bounds grow like (4 - 1/(4g))n for g≥2.

Diameter bounds are tight for the torus case.

Abstract

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.