Injective maps between flip graphs

TL;DR

This paper characterizes injective simplicial maps between flip graphs of surfaces, showing they are mostly induced by subsurface inclusions, extending known results about automorphisms to a broader class of maps.

Contribution

It generalizes the understanding of flip graph maps by proving that injective simplicial maps are mostly induced by subsurface inclusions, extending previous automorphism results.

Findings

Injective simplicial maps are induced by subsurface inclusions in most cases.

Extends Korkmaz--Papadopoulos result from automorphisms to injective maps.

Identifies finitely many exceptions to the general rule.

Abstract

We prove that every injective simplicial map $\mathcal{F}(S) \to \mathcal{F}(S')$ between flip graphs is induced by a subsurface inclusion $S\to S'$, except in finitely many cases. This extends a result of Korkmaz--Papadopoulos which asserts that every automorphism of the flip graph of a surface without boundary is induced by a surface homeomorphism.