Two remarks about multicurve graphs on 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.