Bounds for the number of moves between pants decompositions, and between triangulations

TL;DR

The article proves a universal upper bound on the distance between pants decompositions in the pants graph, depending logarithmically on their intersection i(P,P') and polynomially on the surface's Euler characteristic | chi(S)|. It develops a robust framework using pre-triangulations, train tracks, and the Agol–Hass–Thurston algorithm to translate combinatorial changes into controlled geometric moves, and then applies these results to volumes of maximal cusps and Weil–Petersson geometry. The work also extends to upper bounds for flips/twists between triangulations, polygonal decompositions, spines, and their algorithmic realizations, with concrete consequences for hyperbolic 3-manifolds and Teichmüller theory. Together, these results illuminate the quantitative relationship between surface decompositions, three-manifold geometry, and moduli space metrics, offering practical algorithms and sharp asymptotics. The bounds are shown to be near-optimal up to a surface-dependent factor, and the methods open avenues for further algorithmic applications in low-dimensional topology and geometry.

Abstract

Given two pants decompositions of a compact orientable surface $S$, we give an upper bound for their distance in the pants graph that depends logarithmically on their intersection number and polynomially on the Euler characteristic of $S$. As a consequence, we find an upper bound on the volume of the convex core of a maximal cusp (which is a hyperbolic structures on $S \times \mathbb{R}$ where given pants decompositions of the conformal boundary are pinched to annular cusps). As a further application, we give an upper bound for the Weil--Petersson distance between two points in the Teichmüller space of $S$ in terms of their corresponding short pants decompositions. Similarly, given two one-vertex triangulations of $S$, we give an upper bound for the number of flips and twist maps needed to convert one triangulation into the other. The proofs rely on using pre-triangulations, train tracks, and an algorithm of Agol, Hass, and Thurston.

Figures (28)

• Figure 1: The edges of the pants graph correspond to simple moves (left) and associativity moves (right). In each case, the red curve is replaced with the blue curve, and the remaining curves are unchanged.
• Figure 2: The three types of normal arcs in a triangle.
• Figure 3: The windows $w_i$ and the arcs $\ell_{i,j}$.
• Figure 4: Here $ABC$ and $A'B'C'$ show two triangles of $\mathcal{T}$. Left: The three (blue dashed) arcs $AB$, $AD$, and $EF$ of $\mathcal{T}'$ are normal with respect to $\mathcal{T}$. Right: The two (blue dashed) arcs $A'D'$ and $E'F'$ of $\mathcal{T'}$ are not normal with respect to $\mathcal{T}$.
• Figure 5: Here the two triangles belong to the polygonal decomposition $\mathcal{T}$, and the blue dashed 1-complex is the intersection of the 1-complex $\gamma$ with the corresponding triangles. On the left the restriction of $\gamma$ to the triangle is in normal form with respect to $\mathcal{T}$. On the right $\gamma$ in not in normal form with respect to $\mathcal{T}$ because of either of the two shaded bigons between $\mathcal{T}$ and $\gamma$.

Theorems & Definitions (96)

• Example 1.1
• Theorem 1.2
• Remark 1.3
• Definition 1.5: Pre-triangulation
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Example 2.3: Euclid's algorithm for GCD
• Definition 2.4: Shifted cycle of the AHT algorithm
• Definition 2.5: Normal form for a triangulation