Counting geodesics between surface triangulations

TL;DR

This paper studies the flip-graph of surface triangulations, showing that the number of shortest paths (geodesics) between two triangulations grows exponentially for complex surfaces and polynomially for simpler ones.

Contribution

It characterizes the growth rate of the number of geodesics in the flip-graph based on the surface's topology, providing new insights into its combinatorial structure.

Findings

Exponential growth of geodesics for surfaces with sufficient topology.

Polynomial growth of geodesics for simpler surfaces.

Provides a classification based on surface topology.

Abstract

Given a surface $\Sigma$ equipped with a set $P$ of marked points, we consider the triangulations of $\Sigma$ with vertex set $P$. The flip-graph of $\Sigma$ whose vertices are these triangulations, and whose edges correspond to flipping arcs appears in the study of moduli spaces and mapping class groups. We consider the number of geodesics in the flip-graph of $\Sigma$ between two triangulations as a function of their distance. We show that this number grows exponentially provided the surface has enough topology, and that in the remaining cases the growth is polynomial.