Combinatorial rigidity of arc complexes

Abstract

We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every automorphism is induced by a homeomorphism of the surface. As an application we deduce some rigidity results for the Fomin-Shapiro-Thurston cluster algebra associated to such a surface. Our proofs do not employ any known simplicial rigidity result.

Revised version: 29 pages, 14 figures

References in corpus (2)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs

Cited by in corpus (5)

• On auto-equivalences and complete derived invariants of gentle algebras
• Normal subgroups of mapping class groups and the metaconjecture of Ivanov
• Geometric simplicial embeddings of arc-type graphs
• Automorphisms of the k-curve graph
• Bijections of geodesic lamination space preserving left Hausdorff convergence