Smallest noncyclic quotients of braid and mapping class groups

Kolay, Sudipta¹

¹ School of Mathematics, Georgia Institute of Technology, Atlanta, GA, United States

Geometry & topology, Tome 27 (2023) no. 6, pp. 2479-2496.

Résumé

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.

Détail

DOI : 10.2140/gt.2023.27.2479

Keywords: smallest noncyclic quotients, braid groups, mapping class groups

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