Salter's question on the image of the Burau representation of $B_4$

TL;DR

This paper addresses Salter's question about the image of the Burau representation of B4, providing counterexamples under the assumption of faithfulness and discussing related faithfulness properties.

Contribution

It constructs counterexamples to Salter's question on the Burau representation of B4 and offers new insights into the faithfulness of the representation.

Findings

Counterexamples to Salter's question are constructed assuming faithfulness.

The restriction to the centralizer of a generator is faithful modulo p for all primes p.

A building-theoretic criterion for faithfulness is proposed.

Abstract

In 1974, Birman posed the problem of identifying the conditions under which a matrix with Laurent polynomial entries lies in the image of the Burau representation. Building on this, Salter, in 2021, refined the inquiry to ask whether the central quotient of the Burau image group coincides with the central quotient of a specific subgroup of the unitary group. Assuming the faithfulness of the $B_{4}$ Burau, we solve Salter's question negatively in the case $n=4$ constructing counterexamples. Additionally, we offer two remarks on the faithfulness of the $B_{4}$ Burau. First, we establish that the restriction to the centralizer of a standard generator in $B_{4}$ is faithful modulo $p$ for every prime $p$, extending both Smythe's result in 1979 and Moran's result in 1991. Second, we present a building-theoretic criterion for the faithfulness.