  We have investigated the ground state properties of solid $^4$He with the
Shadow Path Integral Ground State method. This exact T=0 K projector method
allows to describes quantum solids without introducing any a priori equilibrium
position. We have found that the efficiency in computing off-diagonal
properties in the solid phase sensibly improves when the direct sampling of
permutations, in principle not required, is introduced. We have computed the
exact one-body density matrix (obdm) in large commensurate 4He crystal finding
a decreasing condensate fraction with increasing imaginary time of projection,
making our result not conclusive on the presence of Bose-Einstein condensation
in bulk solid 4He. We can only give an upper bound of 2.5 times 10^-8 on the
condensate fraction. We have exploited the SPIGS method to study also 4He
crystal containing grain boundaries by computing the related surface energy and
the obdm along these defects. We have found that also highly symmetrical grain
boundaries have a finite condensate fraction. We have also derived a route for
the estimation of the true equilibrium concentration of vacancies x_v in bulk
T=0 K solid 4He, which is shown to be finite, x_v=0.0014(1) at the melting
density, when computed with the variational shadow wave function technique.
