Counting geodesics between surface triangulations

TL;DR

The paper addresses counting geodesics in flip-graphs of surface triangulations by introducing the key quantity $\Delta_k(\Sigma)$ and proving a sharp dichotomy: for most topologies with sufficient complexity the geodesic count grows exponentially in $k$, while in low-complexity cases such as the cylinder without punctures and the $2$-punctured disk the growth is polynomial. The main technique is to produce strongly convex subgraphs isomorphic to $\mathcal{F}(\Gamma)\times\mathcal{F}(\Gamma)$, using contraction operations and the relation to $\mathcal{F}(\Sigma^*)$, which yields explicit exponential lower bounds. The authors also derive a general upper bound tying $\Delta_k(\Sigma)$ to quantities on $\Sigma^*$, specifically $\Lambda_k(\Sigma^*)$ and $\widetilde{\Delta}_k(\Sigma^*)$, with a power $r=(2\kappa(\Sigma))^{n-b}$. Together, these results illuminate the geodesic complexity of flip-graphs in relation to surface topology, providing both a global exponential regime and precise polynomial bounds in the few remaining simple cases, though two small topologies remain unresolved. The work advances understanding of the combinatorial and geometric structure underlying moduli-type spaces and mapping class group actions via flip-graphs.

Abstract

Given a surface $Σ$ equipped with a set $P$ of marked points, we consider the triangulations of $Σ$ with vertex set $P$. The flip-graph of $Σ$ 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 $Σ$ 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.

Figures (6)

• Figure 1: The behavior of $\Delta_k(\Sigma)$ when $b\leq3$, $g\leq2$ and $p\leq5$. Symbols indicate the behavior of $\Delta_k(\Sigma)$ as $k$ goes to infinity: solid disks indicate exponential behavior, striped disks polynomial behavior, solid circles that $\Delta_k(\Sigma)=1$, and dotted circles represent unsolved cases. Triangles mean that $\mathcal{F}(\Sigma)$ is finite. When $b=0$ and there is no symbol, $\mathcal{F}(\Sigma)$ is empty.
• Figure 2: The flip-graph of a $3$-punctured sphere.
• Figure 3: The contraction of $\alpha$ in a triangulation of $\Sigma$.
• Figure 4: The constructions in Propositions \ref{['PP4.sec.2.prop.1']}, \ref{['PP4.sec.2.prop.2']}, and \ref{['PP4.sec.2.prop.3']}.
• Figure 5: A portion of the flip-graph of a cylinder without punctures and one marked point on each boundary.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 3.1
• Proposition 3.2
• Proposition 3.3