Finite image homomorphisms of the braid group and its generalizations

Abstract

Using totally symmetric sets, Chudnovsky, Kordek, Li, and Partin gave a superexponential lower bound on the cardinality of non-abelian finite quotients of the braid group. In this paper, we develop new techniques using multiple totally symmetric sets to count elements in non-abelian finite quotients of the braid group. Using these techniques, we improve the lower bound found by Chudnovsky et al. We exhibit totally symmetric sets in the virtual and welded braid groups, and use our new techniques to find superexponential bounds for the finite quotients of the virtual and welded braid groups.A guiding question to motivate Theorem A is the following conjecture.Conjecture (Margalit [5]). For n ≥ 5, Σ n is the smallest finite, non-cyclic quotient of B n .

Homomorphisms of commutator subgroups of braid groups

“…(4) Chen and Mukherjea [11] classify homomorphisms from 𝐵 𝑛 to the mapping class group of a surface of genus g ⩽ 𝑛 − 3. (5) Scherich and Verberne [21] improved on the aforementioned lower bound of Chudnovsky, Li, Partin, and the first author.…”