The coarse geometry of hexagon decomposition graphs

TL;DR

The paper introduces two hexagon-decomposition graphs, $\mathcal{H}(\Sigma)$ (topological) and $\mathcal{H}(X)$ (geometric), built from pairs $(\Gamma, \mathcal{A})$ of curves and arcs that decompose a surface into hexagons. The topological variant is shown to be quasi-isometric to the pants graph $\mathcal{P}(\Sigma)$, making it a coarse model for the Weil–Petersson geometry, while the geometric variant adds integer weights on curves via a fixed hyperbolic metric $X$ and is quasi-isometric to the mapping class group $\mathrm{Mod}(X)$ (and to the marking graph by Masur–Minsky). The results hinge on constructing and analyzing flips, curve additions/removals, and weight-update rules, along with a Milnor–Schwarz–type argument for the action of $\mathrm{Mod}(X)$ on $\mathcal{H}(X)$. Together, these graphs provide coarse geometric models for deformation spaces and symmetry groups of surfaces, linking combinatorial, hyperbolic, and mapping class group theories.

Abstract

We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group.

Figures (10)

• Figure 1: The first type of elementary move is on a four-holed sphere
• Figure 2: The second type of elementary move is on a one-holed torus
• Figure 3: The illustration of a flip. Note that the curves of $\Gamma$ the arcs terminate on need not be distinct.
• Figure 4: Arcs bounding embedded cylinders in the non-planar and planar cases
• Figure 5: A hexagon decomposition of the cylinder and its core curve

Theorems & Definitions (29)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Definition 3.2
• Remark 3.3
• Lemma 3.4
• proof
• Theorem 3.5
• proof