Combinatorial complexes associated to surfaces

Shinkle, Emily Suzanne

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

Description

Title

Combinatorial complexes associated to surfaces

Author(s)

Shinkle, Emily Suzanne

Issue Date

2023-04-07

Director of Research (if dissertation) or Advisor (if thesis)

Leininger, Christopher J

Doctoral Committee Chair(s)

Dunfield, Nathan M

Committee Member(s)

• Albin, Pierre
• Sadanand, Chandrika

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• mapping class group
• flip graph
• arc complex
• rigidity
• finite rigid sets
• finite rigidity
• geometric topology
• geometric group theory
• arcs on surfaces
• triangulations of surfaces

Abstract

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.

Graduation Semester

2023-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2023 Emily Shinkle

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