Flip graph and arc complex finite rigidity

TL;DR

This paper demonstrates that the flip graph's finite rigidity implies the arc complex's finite rigidity for surfaces with marked points, including those with boundary, extending previous results in geometric topology.

Contribution

It establishes a link between the finite rigidity of flip graphs and arc complexes, proving finite rigidity for a broad class of surfaces, including those with boundary.

Findings

Finite rigidity of flip graphs implies finite rigidity of arc complexes.

Finite rigidity of arc complexes is proven for surfaces with boundary.

Extends known results to new classes of surfaces.

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 $\lambda:\mathcal{X} \rightarrow \mathcal{C}'$ has a unique extension to an injective simplicial map $\phi:\mathcal{C}\rightarrow \mathcal{C}'$. 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.