Combinatorial complexes associated to surfaces

We study compact, connected, orientable surfaces, sometimes with boundary, and with a positive number of marked points such that there is at least one marked point on any boundary component. For surfaces without boundary and not isomorphic to a 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. Further, we show that for most pairs of surfaces, possibly with boundary, there exists a finite subgraph of the flip graph of the first surface so that any injective homomorphism of this finite subgraph into the flip graph of the second surface can be extended uniquely to an injective homomorphism between the two flip graphs. Combined with a result of Aramayona-Koberda-Parlier, this implies that any such injective homomorphism of this finite set is induced by an embedding of the surfaces.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Shinkle, Emily Suzanne

Contributors dc:contributor

• Leininger, Christopher J
• Dunfield, Nathan M
• Albin, Pierre
• Sadanand, Chandrika

• Mapping Class Group
• Flip Graph
• Arc Complex
• Rigidity
• Finite Rigid Sets
• Finite Rigidity
• Geometric Topology
• Geometric Group Theory
• Arcs On Surfaces
• Triangulations Of Surfaces

Statement dc:rights

• Copyright 2023 Emily Shinkle

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120241

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120241