  We investigate the spectrum of the three-dimensional Dirichlet Laplacian in a
prototypal infinite polyhedral layer, that is formed by three perpendicular
quarter-plane walls of constant width joining each other. Alternatively, this
domain can be viewed as an octant from which another "parallel" octant is
removed. It contains six edges (three convex and three non-convex) and two
corners (one convex and one non-convex). It is a canonical example of
non-smooth conical layer. We name it after Fichera because near its non-convex
corner, it coincides with the famous Fichera cube that illustrates the
interaction between edge and corner singularities. This domain could also be
called an octant layer.
  We show that the essential spectrum of the Laplacian on such a domain is a
half-line and we characterize its minimum as the first eigenvalue of the
two-dimensional Laplacian on a broken guide. By a Born-Oppenheimer type
strategy, we also prove that its discrete spectrum is finite and that a lower
bound is given by the ground state of a special Sturm-Liouville operator. By
finite element computations taking singularities into account, we exhibit
exactly one eigenvalue under the essential spectrum threshold leaving a
relative gap of 3%. We extend these results to a variant of the Fichera layer
with rounded edges (for which we find a very small relative gap of 0.5%), and
to a three-dimensional cross where the three walls are full thickened planes.
