Finite rigid subgraphs of the pants graphs of punctured spheres

TL;DR

This paper establishes a strong finite rigidity property for certain subgraphs of the pants graphs of punctured spheres, showing that embeddings of these subgraphs correspond to embeddings of the underlying surfaces.

Contribution

The authors construct specific finite subgraphs of pants graphs of punctured spheres that exhibit a rigidity property, linking graph embeddings to surface embeddings.

Findings

Finite subgraphs $X_n$ of pants graphs are constructed for $n ext{ } ext{and}$ $m$ punctured spheres.

Any simplicial embedding of $X_n$ into another pants graph is induced by a surface embedding.

The result demonstrates a strong form of finite rigidity in the combinatorial structure of pants graphs.

Abstract

We prove a strong form of finite rigidity for pants graphs of spheres. Specifically, for any $n\geq4$, we construct a finite subgraph $X_n$ of the pants graph $P(S_{0,n})$ of the n-punctured sphere $S_{0,n}$ with the following property. Any simplicial embedding of $X_n$ into any pants graph $P(S_{0,m})$ of a punctured sphere is induced by an embedding $S_{0,n} \to S_{0,m}$.