On a remarkable class of polyhedra in complex hyperbolic space

Dedicated to Gerhard Hochschild on the occasion of his 65th birthday The Selberg, Piatetsky-Shapiro conjecture, now establi-shed 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 w-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 %-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 con-jecture. 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. This paper arose out of an attempt to determine the limits of validity of the Selberg, Piatetsky-Shapiro conjecture on the arith-meticity of lattice subgroups. In 1960 A. Selberg conjectured that apart from some exceptional G, an irreducible noncocompact lattice subgroup Γ of a semi-simple group G is arithmetic (irreducible in the sense that Γ is not commensurable with a direct product of its intersections with factors of G). Later Piatetsky-Shapiro conjec-tured: An irreducible lattice of a semi-simple group G is arithmetic if U-rank G 1. The Selberg, Piatetsky-Shapiro conjecture was settled affirma-tively by G. A. Margoulis in the striking paper that he submitted