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Finite rigid sets in arc complexes

Emily Shinkle

Algebraic & Geometric Topology 20 (2020) 3127–3145
DOI: 10.2140/agt.2020.20.3127

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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arc complex, rigidity, finite rigid, arc, simplicial complex, geometric topology
Mathematical Subject Classification 2010
Primary: 20F38, 20F65, 57M60, 57N05
Received: 7 October 2019
Revised: 20 January 2020
Accepted: 17 February 2020
Published: 8 December 2020
Emily Shinkle
University of Illinois at Urbana-Champaign
Urbana, IL
United States