The Selberg, Piatetsky-Shapiro
conjecture, now established by Margoulis, asserts that an irreducible lattice in a
semi-simple group G is arithmetic if the real rank of G is greater than one.
Arithmetic lattices are known to exist in the real-rank one group SO(n,1), the
motion group of real hyperbolic n-space, for n ≦ 5. These examples due to Makarov
for n = 3 and Vinberg for n ≦ 5 are defined by reflecting certain finite volume
polyhedra in real hyperbolic n-space through their faces. The purpose of the present
paper is to show that there are also nonarithmetic lattices in the real-rank one group
PU(2,1), the group of motions of complex hyperbolic 2-space which can be defined
algebraically and leads to remarkable polyhedra. This serves to help determine the
limits of the Selberg, Piatetsky-Shapiro conjecture. The analysis of these
polyhedra also leads to the first known example of a compact negatively
curved Riemannian space which is not diffeomorphic to a locally symmetric
space.
