We present angle-resolved photoemission measurements for ultrathin In films on Si(111). Depending on the coverage, this system self-organizes into a metallic monolayer with either 4x1 or sqrt7 x sqrt3 symmetry relative to the substrate. Electronically, they behave like ideal one- and two-dimensional electron gases (1DEG and 2DEG), respectively. The 4x1 system has atomic chains of In whose energy bands disperse only parallel to the chains, while for the sqrt7 x sqrt3 system, the dominant reciprocal space features (in both diffraction and bandstructure) resemble a pseudo-square lattice with only weaker secondary features relating to the sqrt7 x sqrt3 periodicity. In both materials the electrons show coupling to the structure. The 1DEG couples strongly to phonons of momentum 2kF, leading to an 8x"2" Peierls-like insulating ground state. The 2DEG appears to be partially stabilized by electron gap formation at the sqrt 7 x sqrt3 zone boundary.