Smallest non-cyclic quotients of braid and mapping class groups

TL;DR

This paper proves that the smallest non-cyclic quotients of braid and mapping class groups are symmetric groups and mod two symplectic groups respectively, providing elementary proofs and confirming longstanding conjectures.

Contribution

It establishes the minimal non-cyclic quotients for braid and mapping class groups without relying on the Bertrand-Chebyshev theorem, confirming conjectures and simplifying proofs.

Findings

Smallest non-cyclic quotient of braid groups is the symmetric group.

Smallest non-cyclic quotient of mapping class groups is the mod two symplectic group.

Provides elementary proofs for these classifications.

Abstract

We show that the smallest non-cyclic quotients of braid groups are symmetric groups, proving a conjecture of Margalit. Moreover we recover results of Artin and Lin about the classification of homomorphisms from braid groups on n strands to symmetric groups on k letters, where k is at most n. Unlike the original proofs, our method does not use the Bertrand-Chebyshev theorem, answering a question of Artin. Similarly for mapping class group of closed orientable surfaces, the smallest non-cyclic quotient is given by the mod two reduction of the symplectic representation. We provide an elementary proof of this result, originally due to Kielak-Pierro, which proves a conjecture of Zimmermann.