Small Quotients of Braid Groups

Abstract

We prove that the symmetric group Sn is the smallest non-cyclic quotient of the braid group Bn for n = 5, 6 and that the alternating group An is the smallest non-trivial quotient of the commutator subgroup B ′ n for n = 5, 6, 7, 8. We also give an improved lower bound on the order of any non-cyclic quotient of Bn.

Finite image homomorphisms of the braid group and its generalizations

“…One main focus of this paper is to understand the non-cyclic quotients of B n . Work by Chudnovsky-Kordek-Li-Partin [5], and more recently by Caplinger-Kordek [4]), proves a lower bound for the size of non-cyclic quotients of B n . In this paper, we provide an improved lower bound for the size of non-cylic quotients of B n by adding a polynomial term to the result of Caplinger-Kordek, as found in Theorem A.…”

“…Chudnovsky, Kordek, Li, and Partin utilized the existence of these totally symmetric sets to determine a necessary condition for the existence of a non-cyclic homomorphism from the braid group into a group [5]. Recently, Caplinger and Kordek obtained a stronger necessary condition than the one found by Chudnovsky-Kordek-Li-Partin [4]. Lemma 3.1 (Caplinger-Kordek).…”

“…When n = 5, 6, and 7, Caplinger and Kordek used the classification of finite groups to conclude that a non-cyclic quotient of B n must be larger than n! [4]. Since Theorem A gives a lower bound on the size of the image of a non-cyclic homomorphism for n > 5, it therefore gives the tightest known lower bound for n ≥ 8.…”

“…The theory of totally symmetric sets was first introduced by Kordek and Margalit when studying homomorphisms from the commutator subgroup of the braid group on n strands to the braid group on n strands [9]. More recently, totally symmetric sets were used by Caplinger-Kordek [4] and Chudnovsky-Kordek-Li-Partin [5] when studying finite quotients of the braid group. Totally symmetric sets are useful for counting arguments since the size of the image of a totally symmetric set under a homomorphism is either equal to 1, or the size of the totally symmetric set.…”

Homomorphisms of commutator subgroups of braid groups

“…(2) Li, Partin, and the first author [16] give upper bounds on the cardinalities of totally symmetric sets in various types of groups. (3) Caplinger and the first author [9] show that the smallest non-abelian finite quotients of 𝐵 5 and 𝐵 6 are the corresponding symmetric groups. (4) Chen and Mukherjea [11] classify homomorphisms from 𝐵 𝑛 to the mapping class group of a surface of genus g ⩽ 𝑛 − 3.…”