Vol. 86, No. 1, 1980

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On a remarkable class of polyhedra in complex hyperbolic space

George Daniel Mostow

Vol. 86 (1980), No. 1, 171–276

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.

Mathematical Subject Classification 2000
Primary: 22E40
Secondary: 30F35, 51M10
Received: 3 August 1979
Published: 1 January 1980
George Daniel Mostow