  Based on the observation that the moduli of a link variable on a cyclic group
modify Connes' distance on this group, we construct several action functionals
for this link variable within the framework of noncommutative geometry. After
solving the equations of motion, we find that one type of action gives
nontrivial vacuum solution for gravity on this cyclic group in a broad range of
coupling constants and that such solutions can be expressed with Chebyshev's
polynomials.
