Flip graph and arc complex finite rigidity

A subcomplex $\mathcal{X}$ of a cell complex $\mathcal{C}$ is called \emph{rigid} with respect to another cell complex ${\mathcal{C}}^{\mathrm{\prime }}$ if every injective simplicial map $\lambda :\mathcal{X}\to {\mathcal{C}}^{\mathrm{\prime }}$ has a unique extension to an injective simplicial map $\phi :\mathcal{C}\to {\mathcal{C}}^{\mathrm{\prime }}$. We say that a cell complex exhibits \emph{finite rigidity} if it contains a finite rigid subcomplex. Given a surface with marked points, its $\mathit{\text{flip}}\mathit{\text{graph}}$ and $\mathit{\text{arc}}\mathit{\text{complex}}$ 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.