Simplicial embeddings between multicurve graphs

TL;DR

This paper investigates the structure of multicurve graphs associated with surfaces, demonstrating that simplicial embeddings correspond to surface embeddings and establishing their rigidity, thus advancing understanding of surface-related graph symmetries.

Contribution

It proves that simplicial embeddings between multicurve graphs are induced by surface embeddings and establishes the rigidity of these graphs.

Findings

Simplicial embeddings correspond to surface embeddings.

Multicurve graphs exhibit rigidity properties.

Results connect graph embeddings with topological surface embeddings.

Abstract

We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by $\pi_1$-injective embeddings of the corresponding surfaces. We also prove the rigidity of the multicurve graphs.