On the subgroup of $B_4$ that contains the kernel of Burau representation
It is known that there are braids $α$ and $β$ in the braid group $B_4$, such that the group $\langle α, β\rangle$ is a fee subgroup \cite{7}, which contains the kernel $K$ of the Burau map $ρ_4 : B_4 \to G L\left(3, \mathbb{Z}[t,t^{-1}]\right)$ \cite{6}, \cite{4}. In this paper we will prove that $K$ is subgroup of $G=\langle τ, Δ\rangle $, where $τ$ and $Δ$ are fourth and square roots of the generator $θ$ of the center $Z$ of the group $B_4$. Consequently, we will write elements of $K$ in terms of $τ^i,~~i=1,2,3$ and $Δ$. Moreover, we will show that the quotient group $G/Z$ is isomorphic to the free product $Z_4 *Z_2$.
math.GR↗