Finite Rigid Sets in Arc Complexes

TL;DR

This paper constructs finite rigid sets within the arc complexes of certain surfaces, enabling the recognition of surface homeomorphisms from arc complex isomorphisms, extending prior results by Irmak and McCarthy.

Contribution

It introduces a method to build finite rigid sets in arc complexes for most surfaces, providing tools for surface classification via arc complex isomorphisms.

Findings

Finite rigid sets are constructed for most surfaces.

Arc complex isomorphisms imply surface homeomorphisms.

An exhaustion of the arc complex by finite rigid sets is provided.

Abstract

For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by finite rigid sets. This extends the results of Irmak--McCarthy.