Spaces of Pants Decompositions for Surfaces of Infinite Type (2010.13169v3)

Abstract: We study the pants complex of surfaces of infinite type. When $S$ is a surface of infinite type, the usual definition of the pants graph $\mathcal{P}(S)$ yields a graph with infinitely many connected-components. In the first part of our paper, we study this disconnected graph. In particular, we show that the extended mapping class group $\mathrm{Mod}(S)$ is isomorphic to a proper subgroup of $\mathrm{Aut}(\mathcal{P})$, in contrast to the finite-type case where $\mathrm{Mod}(S)\cong \mathrm{Aut}(\mathcal{P}(S))$. In the second part of the paper, motivated by the Metaconjecture of Ivanov, we seek to endow $\mathcal{P}(S)$ with additional structure. To this end, we define a coarser topology on $\mathcal{P}(S)$ than the topology inherited from the graph structure. We show that our new space is path-connected, and that its automorphism group is isomorphic to $\mathrm{Mod}(S)$.