Spaces of Pants Decompositions for Surfaces of Infinite Type

TL;DR

This paper investigates the structure of the pants complex for infinite-type surfaces, revealing its disconnected nature, the relationship with the mapping class group, and introducing a new topology that restores connectedness and automorphism group correspondence.

Contribution

It demonstrates that the usual pants graph for infinite-type surfaces is disconnected, and introduces a new topology making the automorphism group isomorphic to the mapping class group.

Findings

The pants graph for infinite-type surfaces has infinitely many connected components.

The extended mapping class group is a proper subgroup of the automorphism group of the pants graph.

A new topology on the pants complex makes it path-connected and its automorphism group isomorphic to the mapping class group.

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 $\mathrm{Mod}(S)$.