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Totally Symmetric Sets

Noah Caplinger, Dan Margalit

Abstract

We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.

Totally Symmetric Sets

Abstract

We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.
Paper Structure (11 sections, 11 theorems, 52 equations)

This paper contains 11 sections, 11 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Totally symmetric sets and the blueprint
  3. Totally symmetric sets and collision implies collapse
  4. Totally symmetric configurations and upper bounds on totally symmetric sets
  5. Application to the symmetric group
  6. Totally symmetric sets in the general linear group
  7. Totally symmetric sets in braid groups
  8. Finite quotients of braid groups and mapping class groups
  9. Braid groups
  10. Mapping class groups
  11. Speculations and representations

Key Result

Lemma 2.1

Let $f : G \to H$ be a homomorphism of groups. If $X \subseteq G$ is a totally symmetric set then $f(X)$ is either a totally symmetric set of cardinality $|X|$ or a singleton.

Theorems & Definitions (13)

  • Lemma 2.1
  • proof
  • Theorem 2.2: Caplinger
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 3.1: Caplinger--Salter
  • Proposition 3.2: Caplinger--Salter
  • Theorem 4.1: Dyer--Grossman
  • Proposition 4.2: Kordek--Margalit
  • Proposition 4.3
  • ...and 3 more