Smallest noncyclic quotients of braid and mapping class groups

Kolay, Sudipta

doi:10.2140/gt.2023.27.2479

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Smallest noncyclic quotients of braid and mapping class groups

Sudipta Kolay

Geometry & Topology 27 (2023) 2479–2496

DOI: 10.2140/gt.2023.27.2479

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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Volume 27, issue 6 (2023)