We study chiral fields [Ui in the group U(N)] on a periodic lattice (Ui=Ui+L), with action S=(1g2)Σl=1LTr(UlUl+1†+Ul†Ul+1), as prototypes for lattice gauge theories [quantum chromodynamics (QCD)] at Nc=∞. Indeed, these chiral chains are equivalent to gauge theories on the surface of an L-faced polyhedron (e.g., L=4 is a tetrahedron, L=6 is the cube, and L=∞ is two-dimensional QCD). The one-link Schwinger-Dyson equation of Brower and Nauenberg, which gives the square of the transfer matrix, is solved exactly for all N. From the large-N solution, we solve exactly the finite chains for L=2, 3, 4, and ∞, on the weak-coupling side of the Gross-Witten singularity, which occurs at β=(g2N)−1=14, 13, π8, and 12, respectively. We carry out weak and strong perturbation expansions at Nc=∞ to estimate the singular part for all L, and to show confinement (as g2N→∞) and asymptotic freedom (g2N→0) in the Migdal β function for QCD. The stability of the location of the Gross-Witten singularity for different-size lattices (L) suggests that QCD at Nc=∞ enjoys this singularity in the transition region from strong to weak coupling.