  We report a systematic first principles study on positron annihilation
parameters in the $\beta$-Ga$_2$O$_3$ lattice and Ga mono-vacancy defects
complemented with orientation-dependent experiments of the Doppler broadening
of the positron-electron annihilation. We find that both the
$\beta$-Ga$_2$O$_3$ lattice and the considered defects exhibit unusually strong
anisotropy in their Doppler broadening signals. This anisotropy is associated
with low symmetry of the $\beta$-Ga$_2$O$_3$ crystal structure that leads to
unusual kind of one-dimensional confinement of positrons even in the
delocalized state in the lattice. In particular, the split Ga vacancies
recently observed by scanning transmission electron microscopy produce
unusually anisotropic positron annihilation signals. We show that in
experiments, the positron annihilation signals in $\beta$-Ga$_2$O$_3$ samples
seem to be often dominated by split Ga vacancies.
