Homomorphisms of commutator subgroups of braid groups

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 non-trivial 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.

On quotients of congruence subgroups of braid groups

“…≤ B n[2] and so B n[4] ∩ B n[2] = B n[4]. Then our desired result follows from the first isomorphism theorem.…”

“…An explicit description of φ is given by viewing (Z/2Z) ( n 2 ) as the free abelian group on the image of the n 2 generators of P B n which were denoted A i,j in Corollary 2.4. The kernel of φ is known to be B n [4] (the main theorem of [2]). Now let φ ℓ be the restriction of φ to the subgroup…”

“…where we have also used that B n [4] where the vertical arrows are ℓ th power maps used to construct isomorphisms in Theorem 3.1 and Theorem 3.2. Then our result follows from an application of the five lemma.…”

“…In exploring the cohomology of the level 4 subgroups, Kordek and Margalit, [4], proved the following Theorem 1.6 (Kordek and Margalit, 2019). For each n,…”

From braid groups to mapping class groups

“…We add in another ingredient called the totally symmetric set of a group, which is useful in studying homomorphisms between groups in general. This idea first appears in [KM19] and already has many applications including papers [CKLP19], [CK20], [LP20]. We will give an introduction of totally symmetric sets in Section 2.…”