Homomorphisms of commutator subgroups of braid groups

TL;DR

This paper classifies all homomorphisms from the commutator subgroup of braid groups to the braid group itself for n ≥ 7, showing they extend to automorphisms, thus answering key open questions.

Contribution

It provides a complete classification of such homomorphisms and introduces the use of totally symmetric sets as a new analytical tool.

Findings

Every nontrivial homomorphism extends to an automorphism.

Complete classification of homomorphisms for n ≥ 7.

Answers four open questions of Vladimir Lin.

Abstract

We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7. In particular, we show that each nontrivial homomorphism extends to an automorphism of the braid group on $n$ strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.