Finite Rigid Sets in Flip Graphs

TL;DR

This paper demonstrates that for most surface pairs, finite subgraphs of flip graphs uniquely determine surface embeddings, linking combinatorial properties of flip graphs to topological surface embeddings.

Contribution

It establishes the existence of finite rigid sets in flip graphs that uniquely determine surface embeddings, extending previous results and providing new insights into flip graph automorphisms.

Findings

Finite subgraphs can be extended to full flip graph homomorphisms.

Injective homomorphisms correspond to surface embeddings.

Includes visual representations of flip graphs.

Abstract

We show that for most pairs of surfaces, there exists a finite subgraph of the flip graph of the first surface so that any injective homomorphism of this finite subgraph into the flip graph of the second surface can be extended uniquely to an injective homomorphism between the two flip graphs. Combined with a result of Aramayona-Koberda-Parlier, this implies that any such injective homomorphism of this finite set is induced by an embedding of the surfaces. We also include images of several flip graphs in an appendix.