  We use density functional theory to calculate the electronic band structures,
cohesive energies, phonon dispersions, and optical absorption spectra of
two-dimensional In$_2$X$_2$ crystals, where X is S, Se, or Te. We identify two
crystalline phases (alpha and beta) of monolayers of hexagonal In$_2$X$_2$, and
show that they are characterized by different sets of Raman-active phonon
modes. We find that these materials are indirect-band-gap semiconductors with a
sombrero-shaped dispersion of holes near the valence-band edge. The latter
feature results in a Lifshitz transition (a change in the Fermi-surface
topology of hole-doped In$_2$X$_2$) at hole concentrations $n_{\rm
S}=6.86\times 10^{13}$ cm$^{-2}$, $n_{\rm Se}=6.20\times 10^{13}$ cm$^{-2}$,
and $n_{\rm Te}=2.86\times 10^{13}$ cm$^{-2}$ for X=S, Se, and Te,
respectively, for alpha-In$_2$X$_2$ and $n_{\rm S}=8.32\times 10^{13}$
cm$^{-2}$, $n_{\rm Se}=6.00\times 10^{13}$ cm$^{-2}$, and $n_{\rm
Te}=8.14\times 10^{13}$ cm$^{-2}$ for beta-In$_2$X$_2$.
