Hatcher-Thurston complex for surfaces with non-planar ends

Abstract

In this paper, for each $k\in \mathbb{N}$, we define a complex $Γ_k(S)$ for an infinite-type surface $S$ with non-planar ends, which serves as an analog of the Hatcher-Thurston complex for the infinite-type setting. We show that $Γ_k(S)$ is connected, simply connected, and that the automorphism group of $Γ_k(S)$ is isomorphic to the extended mapping class group.

Figures (11)

• Figure 1: Constructing the path from $v$ to $w$.
• Figure 2: Splitting the closed path $p$ into triangles.
• Figure 3: Examples of the three cases in proof of Lemma \ref{['lem:radius-1']}.
• Figure 4: Splitting the path $p$ into paths of type $1$, $2$, and $3$.
• Figure 5: The construction in Case 1. Here $q_0$ is a $(a_0,a_1)-$segment, $q_1$ is a $a_1-$segment, and $q_2$ is a $a_2-$segment. The path in red has one less segment than $p$.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 1.1: Theorem 1, hernandez
• Theorem 1.2: Theorem 1.1, hatcher
• Definition 1.3
• Lemma 2.1
• proof
• proof : Proof of first half of Theorem \ref{['thm:simply']}
• Definition 2.2