Finite Quotients of Braid Groups Lily Li Joint Work with Alice - - PowerPoint PPT Presentation

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finite quotients of braid groups
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Finite Quotients of Braid Groups Lily Li Joint Work with Alice - - PowerPoint PPT Presentation

Aug 26, 2022 512 likes •823 views

Finite Quotients of Braid Groups Lily Li Joint Work with Alice Chudnovsky, Caleb Partin, and Kevin Kordek Consider... Given two groups G, H, what are all the homomorphisms between them? Braid Group Braid Group Main Result Finite Quotients

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Finite Quotients of Braid Groups

Lily Li Joint Work with Alice Chudnovsky, Caleb Partin, and Kevin Kordek

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Consider...

Given two groups G, H, what are all the homomorphisms between them?

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Braid Group

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Braid Group

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Main Result

Finite Quotients of Braid Groups

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Fundamental Lemma of Totally Symmetric Sets

The image of a totally symmetric set under a homomorphism is a totally symmetric set.

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Fundamental Lemma of Totally Symmetric Sets

The image of a totally symmetric set under a homomorphism is a totally symmetric set.

BUT IT GETS BETTER

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Fundamental Lemma of Totally Symmetric Sets

The image of a totally symmetric set of size n under a homomorphism is a totally symmetric set of size n or 1.

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Our favorite totally symmetric set

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Our favorite totally symmetric set Image of S: also totally symmetric set

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Our favorite totally symmetric set Image of S: also totally symmetric set

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The image of S is either...

singleton Then the map is cyclic

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The image of S is either...

a singleton Then the map is cyclic not a singleton

The stabilizer of S in G Its image is large: contains a copy of Sn

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Case 2: Not singleton

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Case 2: Not singleton

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Case 2: Not singleton

Stabilizer of the totally symmetric set

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Case 2: Not singleton

Stabilizer of the totally symmetric set Size of the totally symmetric set

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Case 2: Not singleton

All the elements that permute S Elements inducing trivial permutation

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Case 2: Not singleton

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Bounds on Sizes of Totally Symmetric Sets