Two remarks about multicurve graphs on infinite-type surfaces

TL;DR

This paper investigates properties of two multicurve graphs on infinite-type surfaces, proving the finiteness of one graph's diameter and characterizing the automorphism group of the other as the extended mapping class group.

Contribution

It extends known results to infinite-type surfaces, showing finiteness of the diameter of one graph and identifying automorphisms of another with the extended mapping class group.

Findings

$ ext{diameter}( ext{}\mathcal{G}_{ ext{infty}}(S) ext{)}$ is finite

Automorphism group of $ ext{ }\mathcal{G}_{0}(S)$ equals the extended mapping class group

Extends finite-type surface results to infinite-type surfaces

Abstract

After Fossas-Parlier, we consider two graphs $\mathcal{G}_{0}(S)$ and $\mathcal{G}_{\infty}(S)$, constructed from multicurves on connected, orientable surfaces of infinite-type. Our first result asserts that $\mathcal{G}_{\infty}(S)$ has finite diameter, which extends a result of Fossas-Parlier. Next, we prove that the group of (label-preserving) automorphisms of $\mathcal{G}_{0}(S)$ is the extended mapping class group of $S$, which may be regarded as an infinite-type analog of a theorem of Margalit about pants complexes.