Automorphism groups of simplicial complexes of infinite-type surfaces

Abstract

Let $S$ be an orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex $\mathcal{C}(S)$, the nonseparating curve complex $\mathcal{N}(S)$, and the Schmutz graph $\mathcal{G}(S)$ of $S$. When all topological ends of $S$ carry genus, we show that all elements in the automorphism groups $\operatorname{Aut}(\mathcal{C}(S))$, $\operatorname{Aut}(\mathcal{N}(S))$, and $\operatorname{Aut}(\mathcal{G}(S))$ are geometric, i.e., these groups are naturally isomorphic to the extended mapping class group $\operatorname{MCG}^{*}(S)$ of the infinite surface $S$. Finally, we study rigidity phenomena within $\operatorname{Aut}(\mathcal{C}(S))$ and $\operatorname{Aut}(\mathcal{N}(S))$.