Disintegrating the curve complex

TL;DR

This paper develops a natural tower of hyperbolic graphs ${\mathcal C}_k(\Sigma)$ that interpolates between the curve graph ${\mathcal C}(\Sigma)$ and a bottom graph ${\mathcal C}_0(\Sigma)$ that is a quasi-tree. Each step ${\mathcal C}_{k+1}(\Sigma)\to{\mathcal C}_k(\Sigma)$ is 1-Lipschitz with coarsely surjective, alignment-preserving behavior and has coarse fibers that are quasi-trees; these fibers, together with the lifting of train-track dynamics, yield hyperbolicity and finite asymptotic dimension bounds ${\rm asdim}{\mathcal C}_k(\Sigma)\le k+1$ and hence ${\rm asdim}{\mathcal C}(\Sigma)\le m(\Sigma)+1$, where $m(\Sigma)$ is the maximal index of a filling train track. The authors classify boundary points as ending laminations of index ≤ $k$, prove acylindricity of the mapping class group action on each level, and give a sharp loxodromic classification depending on lamination index. Collectively, these results provide a detailed hierarchical picture of the curve graph’s geometry, dynamics, and coarse invariants, with the tower offering explicit tools to study asymptotic dimensions and boundary identifications.

Abstract

We study a finite sequence of graphs, beginning with the curve graph and ending with a graph quasi-isometric to a tree. There is a Lipschitz map from one graph in the sequence to the next. This sequence was first introduced by Hamenstädt. We prove (as conjectured by Hamenstädt) that the graphs in this sequence are hyperbolic and that the coarse fibers of the maps in the sequence are quasi-trees. This gives an upper bound on the asymptotic dimension of each graph in the sequence and as a result, an upper bound on the asymptotic dimension of the curve graph. Additionally, we show that the action of the mapping class group on each graph in the sequence is acylindrical, and classify the boundary and actions of individual mapping classes for each graph in the sequence.

Figures (9)

• Figure 1: Constructing a train track with tangent lines avoiding a fixed slope in the range $(-9,-1/9)$.
• Figure 2: Surgeries to obtain a curve cutting off corners. The vertical arc is the side $s$ of $\tau$ whereas the horizontal arc is the constructed arc $a$ cutting off corners.
• Figure 3: The tracks on the bottom are obtained from the track on the top by a left, central, and right split, respectively.
• Figure 4: Constructing a path of curves from $\alpha$ to $\beta$. The curve $\alpha$ is a vertex cycle of $\tau"$ while $\beta$ is a vertex cycle of $\tau"'$. Each edge of the tripod represents a split and the tracks along the leg from $\tau'$ to $\tau"$ carry $\alpha$ but not $\beta$ while the tracks from $\tau'$ to $\tau"'$ carry $\beta$ but not $\alpha$. The blue path is the path $p_{\alpha,\beta}$ from $\alpha$ to $\beta$, obtained by taking vertex cycles of tracks along two legs of the tripod.
• Figure 5: Diagonal extensions push forward under folds. The track on the left is $\tau$ and the track on the right is $\sigma$. The short branches are branches of the diagonal extensions.

Theorems & Definitions (143)

• Corollary 1.1
• Proposition 2.1: BB, Penner
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Definition 2.4
• Theorem 2.5
• Lemma 3.1
• proof