Edge Preserving Maps of the Curve Graphs in Low Genus

Abstract

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components. Let $\mathcal{C}(R)$ be the curve graph of $R$. We prove that if $g=0, n \geq 5$ or $g=1, n \geq 3$, and $\lambda : \mathcal{C}(R) \rightarrow\mathcal{C}(R)$ is an edge preserving map, then $\lambda$ is induced by a homeomorphism of $R$, and this homeomorphism is unique up to isotopy.