mapping class groups of infinite-type surfacessamjfree/presentationmathfest2017.p… · a primer on...
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Mapping Class Groups of Infinite-Type Surfaces
Santana Afton and Samuel Freedman
The College of William & Mary, University of Michigan
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Outline
▪ Finite-Type Surfaces
▪ Infinite-Type Surfaces
▪ Symmetries
▪ Symmetries of Surfaces
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Finite-Type Surfaces
Sphere Torus 2-holed Torus
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Infinite-Type Surfaces
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Infinite-Type Surfaces
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Infinite-Type Surfaces
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Infinite-Type Surfaces
The Flute Surface
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Infinite-Type Surfaces
The Flute Surface The Blooming Cantor Tree
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What is Symmetry?
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What is Symmetry?
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Mapping Classes
Rotation
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Mapping Classes
Dehn Twist
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Mapping Classes
Handle Shift
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Mapping Class Group
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Mapping Class Group: History
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Mapping Class Group: History
(Dehn)
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Mapping Class Group: History
(Patel-Vlamis)
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Mapping Class Group: History
(AF)
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Classes of Handle Shifts
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One-Ended Handle Shifts
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Handle Shifts - Relations
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Basmajian, Ara. "Hyperbolic structures for surfaces of infinite type." Transactions of the American Mathematical Society 336.1 (1993): 421-444.
Farb, Benson, and Dan Margalit. A Primer on Mapping Class Groups (PMS-49). Princeton University Press, 2011.
Patel, Priyam, and Nicholas G. Vlamis. "Algebraic and topological properties of big mapping class groups." arXiv preprint arXiv:1703.02665 (2017).
Richards, Ian. "On the classification of noncompact surfaces." Transactions of the American Mathematical Society 106.2 (1963): 259-269.
Acknowledgements and CitationsThanks to:
Professor Dan Margalit Justin Lanier
Dr. Nick Vlamis
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Questions?
Infinite jail cell window
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What is the Loch Ness Monster?
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Stacks of Waffles