Hatcher-Thurston complex for surfaces with non-planar ends

TL;DR

This paper introduces a new complex for infinite-type surfaces with non-planar ends, demonstrating its connectivity, simple connectivity, and automorphism group isomorphic to the extended mapping class group.

Contribution

It defines the complex mma_k(S) for infinite-type surfaces and proves its key topological and algebraic properties, extending the Hatcher-Thurston complex to an infinite setting.

Findings

mma_k(S) is connected

mma_k(S) is simply connected

Automorphism group of mma_k(S) is isomorphic to the extended mapping class group

Abstract

In this paper, for each $k\in \mathbb{N}$, we define a complex $\Gamma_k(S)$ 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 $\Gamma_k(S)$ is connected, simply connected, and that the automorphism group of $\Gamma_k(S)$ is isomorphic to the extended mapping class group.