Finite Rigid Sets in Arc Complexes

DOI10.2140/AGT.2020.20.3127arXiv1909.08762MaRDI QIDQ6325596

Publication date: 18 September 2019

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.

Mathematics Subject Classification ID

Geometric group theory (20F65) Other groups related to topology or analysis (20F38) Group actions on manifolds and cell complexes in low dimensions (57M60) 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) (57K20)

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