Homomorphisms between braid groups

Abstract

We give a complete classification of homomorphisms from the braid group on $n$ strands to the braid group on $2n$ strands when $n$ is at least 5. We also classify endomorphisms of the braid group on 4 strands, as well as homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $2n-5$ strands. Our classifications suggest a recursive classification of homomorphisms between any braid groups. We also give a simple, geometric proof of a theorem of Lin that highly constrains the holomorphic maps that may exist between spaces of monic, square-free polynomials of two given degrees.

40 pages, 3 figures

References in corpus (4)

• Automorphisms of surface braid groups
• Braids and Permutations
• Homomorphisms of commutator subgroups of braid groups
• Normal subgroups of the braid group and the metaconjecture of Ivanov

Cited by in corpus (4)

• Homomorphisms of commutator subgroups of braid groups
• From braid groups to mapping class groups
• Lower central series, surface braid groups, surjections and permutations
• Smallest nontrivial quotients of the commutator subgroup of braid groups