  We use a geometric approach to show that the reduced Burau representation
specialized at roots of unity has another incarnation as the monodromy
representation of a moduli space of Euclidean cone metrics on the sphere, as
described by Thurston. Using the theory of orbifolds, we leverage this
connection to identify the kernels of these specializations in some cases,
partially addressing a conjecture of Squier. The 4-strand case is the last case
where the faithfulness question for the Burau representation is unknown, a
question that is related e.g. to the question of whether the Jones polynomial
detects the unknot. Our results allow us to place the kernel of this
representation in the intersection of several topologically natural subgroups
of $B_4$.
