The group determinant was studied during the development of
representation theory. In modern language, given a finite group G, its
group determinant is the representation determinant of the right regular
representation of G. Let Y be a hyperbolic 3-manifold with 1 torus
cusp. Suppose that Y has a finite regular cover with covering group G.
We relate the group determinant to the determinant of a matrix that
describes the space of non-separating surfaces with boundary in this
finite cover. As a consequence we obtain that any such manifold Y has
infinitely many fillings that are virtually Haken. This is joint work
with Daryl Cooper.