The Alexander Method for Infinite-Type Surfaces

Abstract

We prove that for any infinite-type orientable surface $S$, there exists a collection of essential curves $\Gamma$ in $S$ such that any homeomorphism that preserves the isotopy classes of the elements of $\Gamma$ is isotopic to the identity. The collection $\Gamma$ is countable and has an infinite complement in $\mathcal{C}\left(S\right)$, the curve complex of $S$. As a consequence, we obtain that the natural action of the extended mapping class group of $S$ on $\mathcal{C}\left(S\right)$ is faithful.