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

Abstract

Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g\ge 2$, $n\ge 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{H}\mathcal{T}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $\lambda :\mathcal{N}(R)\to \mathcal{N}(R)$ is an edge-preserving map, then $\lambda$ is induced by a homeomorphism of $R$. We prove that if $\theta :\mathcal{C}(R)\to \mathcal{C}(R)$ is an edge-preserving map, then $\theta$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed and $\tau :\mathcal{H}\mathcal{T}(R)\to \mathcal{H}\mathcal{T}(R)$ 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 $(g,n)\ne (2,0)$.

arXiv comment38 pages, 25 figures. Changed the statement of Theorem 1.3 and gave more explanation in some proofs