  We report density-functional-theory calculations of the electronic band
structures and optical absorption spectra of two-dimensional crystals of
Ga$_2$X$_2$ (X=S, Se, and Te). Our calculations show that all three
two-dimensional materials are dynamically stable indirect-band-gap
semiconductors with a Mexican-hat dispersion of holes near the top of the
valence band. We predict the existence of Lifshitz transitions---changes in the
Fermi-surface topology of hole-doped Ga$_2$X$_2$---at hole concentrations
$n_{S}=7.96\times 10^{13}$ cm$^{-2}$, $n_{Se}=6.13\times 10^{13}$ cm$^{-2}$,
and $n_{Te}=3.54\times 10^{13}$ cm$^{-2}$.
