Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Flip graph and arc complex finite rigidity

Chandrika Sadanand, Emily Shinkle

TL;DR

The paper addresses finite rigidity for combinatorial models of surfaces, focusing on the arc complex $\mathcal{A}(S)$ and the flip graph $\mathcal{F}(S)$. It leverages a natural duality embedding of $\mathcal{F}(S)$ into $\mathcal{A}(S)$ to transfer finite rigidity from the flip graph to the arc complex, yielding a new finite rigidity result for $\mathcal{A}(S)$ that extends to surfaces with boundary. The main contribution is a constructive bridge: a finite rigid subgraph $\mathcal{X}_{\mathcal{F}}\subseteq \mathcal{F}(S)$ gives rise to a finite rigid subcomplex $\mathcal{X}_{\mathcal{A}}=\cup_{T\in \mathcal{X}_{\mathcal{F}}} \pi(T)$ in $\mathcal{A}(S)$, ensuring unique extensions of injective maps to homeomorphisms. This provides a broad, unified method to establish arc complex rigidity beyond previously known cases and highlights limitations in the converse direction.

Abstract

A subcomplex $\mathcal{X}$ of a cell complex $\mathcal{C}$ is called \emph{rigid} with respect to another cell complex $\mathcal{C}'$ if every injective simplicial map $λ:\mathcal{X} \rightarrow \mathcal{C}'$ has a unique extension to an injective simplicial map $φ:\mathcal{C}\rightarrow \mathcal{C}'$. We say that a cell complex exhibits \emph{finite rigidity} if it contains a finite rigid subcomplex. Given a surface with marked points, its \textit{flip graph} and \textit{arc complex} are simplicial complexes indexing the triangulations and the arcs between marked points, respectively. In this paper, we leverage the fact that the flip graph can be embedded in the arc complex as its dual to show that finite rigidity of the flip graph implies finite rigidity of the arc complex. Thus, a recent result of the second author on the finite rigidity of the flip graph implies finite rigidity of the arc complex for a broad class of surfaces. Notably, this includes surfaces with boundary -- a setting where finite rigidity of the arc complex was previously unknown.

Flip graph and arc complex finite rigidity

TL;DR

The paper addresses finite rigidity for combinatorial models of surfaces, focusing on the arc complex and the flip graph . It leverages a natural duality embedding of into to transfer finite rigidity from the flip graph to the arc complex, yielding a new finite rigidity result for that extends to surfaces with boundary. The main contribution is a constructive bridge: a finite rigid subgraph gives rise to a finite rigid subcomplex in , ensuring unique extensions of injective maps to homeomorphisms. This provides a broad, unified method to establish arc complex rigidity beyond previously known cases and highlights limitations in the converse direction.

Abstract

A subcomplex of a cell complex is called \emph{rigid} with respect to another cell complex if every injective simplicial map has a unique extension to an injective simplicial map . We say that a cell complex exhibits \emph{finite rigidity} if it contains a finite rigid subcomplex. Given a surface with marked points, its \textit{flip graph} and \textit{arc complex} are simplicial complexes indexing the triangulations and the arcs between marked points, respectively. In this paper, we leverage the fact that the flip graph can be embedded in the arc complex as its dual to show that finite rigidity of the flip graph implies finite rigidity of the arc complex. Thus, a recent result of the second author on the finite rigidity of the flip graph implies finite rigidity of the arc complex for a broad class of surfaces. Notably, this includes surfaces with boundary -- a setting where finite rigidity of the arc complex was previously unknown.
Paper Structure (6 sections, 4 theorems, 1 equation)

This paper contains 6 sections, 4 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Background
  4. Duality and finite $\mathcal{A}(S)$ rigidity for surfaces with boundary
  5. Proofs
  6. Discussion

Key Result

Theorem 2.1

Let $S=S_{g,n}$ where $S\not \cong S_{0,n}$ for $n\leq 3$ or $S_{1,1}$. There exists a finite subcomplex $\mathcal{X}\subseteq \mathcal{A}(S)$, such that for any $S'$ with $d(S) \geq d(S')$, and for any locally injective simplicial map $\lambda:\mathcal{X}\rightarrow \mathcal{A}(S')$, there exists a

Theorems & Definitions (6)

  • Theorem 2.1: Arc complex finite rigidity
  • Theorem 2.2: Flip graph finite rigidity
  • Theorem 3.1: Flip graph finite rigidity implies arc complex finite rigidity
  • Corollary 3.2: Arc complex finite rigidity extended
  • proof : Proof of Theorem \ref{['thm: new result']}
  • proof : Proof of Corollary \ref{['cor: main result']}