Classification of the Discrete Real Specializations of the Burau Representation of $B_3$

TL;DR

This paper classifies the discrete real specializations of the Burau representation of B_3 by analyzing hyperbolic isometries and provides examples of unfaithful specializations that are faithful on subgroups.

Contribution

It offers a complete classification of real specializations of the Burau representation of B_3 and presents new examples of unfaithful specializations with subgroup faithfulness.

Findings

Classification of discrete real specializations of B_3

Example of unfaithful specialization of B_4 that is faithful on B_3

Examples of unfaithful specializations of B_3

Abstract

This classification is found by analyzing the action of a normal subgroup of $B_3$ as hyperbolic isometries. This paper gives an example of an unfaithful specialization of the Burau representation on $B_4$ that is faithful when restricted to $B_3$, as well as examples of unfaithful specializations of $B_3$.