On The Faithfulness of the extension of Lawrence-Krammer representation of the group of conjugating automorphisms C_3

TL;DR

This paper investigates whether a specific extension of the Lawrence-Krammer representation of the conjugating automorphism group C_3 is faithful, addressing an open question for the case n=3.

Contribution

The study explores the faithfulness of the extended Lawrence-Krammer representation for C_3, providing insights into its properties and potential faithfulness.

Findings

Analysis of the representation's properties for n=3

Progress towards understanding faithfulness in this case

Open questions remaining for full faithfulness proof

Abstract

Let $C_n$ be the group of conjugating automorphisms. We study the representation $\rho$ of $C_n$, an extension of Lawrence-Krammer representation of the braid group $B_n$, defined by Valerij G. Bardakov. As Bardakov proved that the representation $\rho$ is unfaithful for $n \geq 5$, the cases $n=3,4$ remain open. In our work, we make attempts towards the faithfulness of $\rho$ in the case $n=3$.