  We consider the transmission problem for the Laplace equation on an infinite
three-dimensional wedge, determining the complex parameters for which the
problem is well-posed, and characterizing the infinite multiplicity nature of
the spectrum. This is carried out in two formulations leading to rather
different spectral pictures. One formulation is in terms of square integrable
boundary data, the other is in terms of finite energy solutions. We use the
layer potential method, which requires the harmonic analysis of a
non-commutative non-unimodular group associated with the wedge.
