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We report density-functional-theory calculations of the electronic bandstructures and optical absorption spectra of two-dimensional crystals ofGa$_2$X$_2$ (X=S, Se, and Te). Our calculations show that all threetwo-dimensional materials are dynamically stable indirect-band-gapsemiconductors with a Mexican-hat dispersion of holes near the top of thevalence band. We predict the existence of Lifshitz transitions---changes in theFermi-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$^{-2}$.

Band Structure and Optical Transitions in Atomic Layers of Hexagonal Gallium Chalcogenides