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 and McCarthy (Turkish J. Math. 34 (2010)
339–354).
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