Flip graph and arc complex finite rigidity

Abstract

A subcomplex $X$ of a cell complex $C$ is called rigid with respect to another cell complex $C^{\prime }$ if every injective simplicial map $\lambda :X\to C^{\prime }$ has a unique extension to an injective simplicial map $\varphi :C\to C^{\prime }$. We say that a cell complex exhibits finite rigidity if it contains a finite, rigid subcomplex. Given a surface with marked points, its flip graph and 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. We further show that these arc complexes admit exhaustions by finite rigid sets, which was shown to be an important component in the proof of many interesting model-theoretic properties of simplicial complexes associated to surfaces in a recent work of de la Nuez Gonzalez-Disarlo-Koberda.