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

TL;DR

This paper investigates a specific representation of the conjugating automorphism group $C_n$, extending the Lawrence-Krammer representation, and proves it is unfaithful for $n=3$ and $n=4$, completing previous open cases.

Contribution

The paper proves the unfaithfulness of the extended Lawrence-Krammer representation for $C_n$ when $n=3$ and $n=4$, resolving open cases.

Findings

The representation $ ho$ is unfaithful for $n=3$.

The representation $ ho$ is unfaithful for $n=4.

Abstract

Let $C_n$ be the group of conjugating automorphisms. Valerij G. Bardakov defined a representation $\rho$ of $C_n$, which is an extension of Lawrence-Krammer representation of the braid group $B_n$. Bardakov proved that the representation $\rho$ 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 $\rho$ in the case $n=3$. In this work, we prove that $\rho$ is unfaithful in the both cases $n=3$ and $n=4$.