Homomorphisms of commutator subgroups of braid groups
Kevin Kordek, Dan Margalit
Abstract
We give a complete classification of homomorphisms from the commutator subgroup of the braid group on n $n$ strands to the braid group on n $n$ strands when n $n$ is at least 7. In particular, we show that each non-trivial homomorphism extends to an automorphism of the braid group on n $n$ strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.
Topics & Concepts
Braid groupHomomorphismCommutator subgroupMathematicsBraidCommutatorCombinatoricsAutomorphismBraid theoryPure mathematicsGroup (periodic table)Discrete mathematicsAlgebra over a fieldNormal subgroupPhysicsQuantum mechanicsLie conformal algebraComposite materialMaterials scienceGeometric and Algebraic TopologyAdvanced Algebra and GeometryFinite Group Theory Research