The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms $C_n$ in the cases $n=3$ and $n=4$

Abstract

Let $C_n$ be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation $ρ$ of $C_n$, which is an extension of Lawrence-Krammer representation of the braid group $B_n$. Bardakov proved that the representation $ρ$ is unfaithful for $n \geq 5$. The cases $n=3,4$ remain open. M. N. Nasser and M. N. Abdulrahim made attempts towards the faithfulness of $ρ$ in the case $n=3$. In this work, we prove that $ρ$ is unfaithful in the both cases $n=3$ and $n=4$.