We use density functional theory to calculate the electronic band structures, cohesive energies, phonon dispersions, and optical absorption spectra of two-dimensional In sub(2) X sub(2) crystals, where X is S, Se, or Te. We identify two crystalline phases ( alpha and beta ) of monolayers of hexagonal In sub(2) X sub(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 sub(2) X sub(2)) at hole concentrations n sub(S) = 6.86 x 10 super(13) cm super(-2), n sub(Se) = 6.20 x 10 super(13) cm super(-2) and n sub(Te) = 2.86 x 10 super(13) cm super(-2) for X=S, Se, and Te, respectively, for alpha -In sub(2) X sub(2) and n sub(S) = 8.32 x 10 super(13) cm super(-2), n sub(Se) = 6.00 x 10 super(13) cm super(-2) and n sub(Te) = 8.14 x 10 super(13) cm super(-2) for beta -In sub(2) X sub(2).