Homomorphisms of commutator subgroups of braid groups with small number of strings

Abstract

For any $n$, we describe all endomorphisms of the braid group $B_n$ and of its commutator subgroup $B_n^{\prime }$, as well as all homomorphisms $B_n^{\prime }\to B_n$. These results are new only for small $n$ because endomorphisms of $B_n$ are already described by Castel for $n\ge 6$, and homomorphisms $B_n^{\prime }\to B_n$ and endomorphisms of $B_n^{\prime }$ are already described by Kordek and Margalit for $n\ge 7$. We use very different approaches for $n=4$ and for $n\ge 5$.

13 pages, 4 figures. There is an error in the proof of Lemma 3.2 in Version 1. This error is corrected in Version 2

References in corpus (2)

• Braids and Permutations
• Homomorphisms of commutator subgroups of braid groups

Cited by in corpus (2)

• Endomorphisms of Artin groups of type $B_n$
• Endomorphisms of Artin groups of type ${\tilde{A}}_n$