[Paper Review] The end of the curve complex

Saul Schleimer|ArXiv.org|Aug 21, 2006

Geometric and Algebraic Topology3 references4 citations

TL;DR

This paper proves that for a surface $S_{g,1}$ of genus $g \geq 2$ with one boundary component, the curve complex remains connected even after removing a ball of radius $r$ around any vertex $\omega$. The proof uses subsurface projections, pseudo-Anosov mapping classes, and fibered structures over the punctured surface to construct paths avoiding the ball, establishing that the curve complex has a single end. This resolves a question of Masur and provides evidence for the connectedness of the Gromov boundary.

ABSTRACT

Suppose that S is a surface of genus two or more, with exactly one boundary component. Then the curve complex of S has one end.

Motivation & Objective

• To prove that the curve complex of a surface $S_{g,1}$ with $g \geq 2$ remains connected after removing a ball of radius $r$ around any vertex $\omega$.
• To resolve a question posed by Howard Masur regarding the connectivity of the curve complex outside large balls.
• To provide evidence for the connectedness of the Gromov boundary of the curve complex, as conjectured by P. Storm.
• To show that the curve complex is not quasi-isometric to a tree, answering a question of Bell and Fujiwara in the negative.

Proposed method

• Define the curve complex $\mathcal{C}(S)$ for $S = S_{g,1}$ with vertices as isotopy classes of essential, non-peripheral simple closed curves.
• Use subsurface projection maps $\pi_X$ to relate distances in sub-surfaces to distances in the full complex.
• Construct a fiber $\mathcal{F}_\tau = \rho^{-1}(\tau)$ over a vertex $\tau$ in $\mathcal{C}(S)$, where $\rho$ collapses the boundary to a point.
• Apply Kra’s theorem to show that certain Dehn twists act as pseudo-Anosov maps on subsurfaces, ensuring infinite diameter orbits.
• Use the Bounded Geodesic Image Theorem and geodesic properties to ensure that certain geodesics in $\mathcal{C}(\dot{S})$ must pass through vertices missing a given subsurface.
• Construct paths from any two distant vertices to a common fiber $\mathcal{F}_\tau$ while avoiding the ball $B(\omega, r)$, relying on the dynamics of pseudo-Anosov maps.

Experimental results

Research questions

• RQ1 Does the curve complex of $S_{g,1}$ with $g \geq 2$ remain connected when a ball of radius $r$ is removed from any vertex?
• RQ2 Is the Gromov boundary of the curve complex connected, as suggested by the one-endedness of the complex?
• RQ3 Can the curve complex be quasi-isometric to a tree, given its hyperbolicity and infinite diameter?
• RQ4 How do subsurface projections and pseudo-Anosov dynamics constrain geodesics in the curve complex?
• RQ5 What is the role of the boundary-merging map $\rho: \mathcal{C}(\dot{S}) \to \mathcal{C}(S)$ in preserving connectivity and fiber structure?

Key findings

• For any vertex $\omega$ and radius $r$, the subcomplex $\mathcal{C}^0(\dot{S}) \setminus B(\omega, r)$ is connected, proving the curve complex of $S_{g,1}$ has one end.
• The curve complex is not quasi-isometric to a tree, as it contains non-trivial higher-dimensional connectivity beyond trees.
• The Gromov boundary of $\mathcal{C}(S)$ is not ruled out as connected, as the one-endedness of the complex supports this possibility.
• The fiber $\mathcal{F}_\tau$ over any $\tau \in \mathcal{C}(S)$ is connected, and fibers are preserved under the action of the fundamental group of $S$.
• For any $\alpha, \beta$ outside $B(\omega, 3r)$, there exist paths connecting them to a common fiber $\mathcal{F}_\tau$ while avoiding $B(\omega, r)$, using pseudo-Anosov dynamics.
• The existence of a pseudo-Anosov map on a subsurface $\dot{X} = \dot{S} \setminus \alpha$ ensures that geodesics from $\sigma$ to $\sigma_n = \psi_\gamma^n(\sigma)$ must pass through a vertex missing $\dot{X}$, enabling path construction.