Edge Preserving Maps of the Nonseparating Curve Graphs, Curve Graphs and Rectangle Preserving Maps of the Hatcher-Thurston Graphs

TL;DR

This paper demonstrates that certain edge-preserving and rectangle-preserving maps on various curve and Hatcher-Thurston graphs of a surface are induced by homeomorphisms, establishing a strong link between graph symmetries and surface homeomorphisms.

Contribution

It proves that edge-preserving maps on nonseparating and curve graphs, as well as rectangle-preserving maps on Hatcher-Thurston graphs, correspond to homeomorphisms of the surface, with uniqueness up to isotopy.

Findings

Edge-preserving maps on nonseparating curve graphs are induced by surface homeomorphisms.

Edge-preserving maps on curve graphs are induced by surface homeomorphisms.

Rectangle-preserving maps on Hatcher-Thurston graphs are induced by surface homeomorphisms.

Abstract

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $\lambda : \mathcal{N}(R) \rightarrow\mathcal{N}(R)$ is an edge-preserving map, then $\lambda$ is induced by a homeomorphism of $R$. We prove that if $\theta : \mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is an edge-preserving map, then $\theta$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed and $\tau: \mathcal{HT}(R) \rightarrow\mathcal{HT}(R)$ is a rectangle preserving map, then $\tau$ is induced by a homeomorphism of $R$. We also prove that these homeomorphisms are unique up to isotopy when $(g, n) \neq (2, 0)$.