Finite rigid sets in arc complexes

Shinkle, Emily

doi:10.2140/agt.2020.20.3127

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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

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

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Keywords

arc complex, rigidity, finite rigid, arc, simplicial complex, geometric topology

Mathematical Subject Classification 2010

Primary: 20F38, 20F65, 57M60, 57N05

References

Publication

Received: 7 October 2019

Revised: 20 January 2020

Accepted: 17 February 2020

Published: 8 December 2020

Authors

Emily Shinkle
University of Illinois at Urbana-Champaign Urbana, IL United States

Volume 20, issue 6 (2020)