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Smallest noncyclic quotients of braid and mapping class groups

Sudipta Kolay

Geometry & Topology 27 (2023) 2479–2496
Abstract

We show that the smallest noncyclic 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 noncyclic quotient is given by the mod two reduction of the symplectic representation. We provide an elementary proof of this result, originally due to Kielak and Pierro, which proves a conjecture of Zimmermann.

Keywords
smallest noncyclic quotients, braid groups, mapping class groups
Mathematical Subject Classification
Primary: 20F36, 20F65, 57K20
References
Publication
Received: 19 July 2021
Revised: 5 October 2021
Accepted: 30 October 2021
Published: 25 August 2023
Proposed: Mladen Bestvina
Seconded: David Fisher, Benson Farb
Authors
Sudipta Kolay
School of Mathematics
Georgia Institute of Technology
Atlanta, GA
United States

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