Homomorphisms of commutator subgroups of braid groups
Kevin Kordek, Dan Margalit

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 strands to the braid group on strands when is at least 7. In particular, we show that each nontrivial homomorphism extends to an automorphism of the braid group on strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.
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