Hausdorff dimension of limit sets for spherical CR manifolds

Let M(2n+1) (n greater than or equal to 1) be a compact, spherical CR manifold. Suppose (M) over tilde(2n+1) is its universal cover and Phi:(M) over tilde(2n+1) --> S-2n+1 is on injective CR developing map, where S-2n+1 is the standard unit sphere in the complex (n+1)-space C-n+1, then M(2n+1) is of the quotient form Omega/Gamma, where Omega is a simply connected open set in S-2n+1, and Gamma a complex Klein group acting on Omega properly discontinuously. In this paper, we show that if the CR Yamabe invariant of M(2n+1) is positive, then the Carnot Hausdorff dimension of the limit set of Gamma is bounded above by n . s(M(2n+1)), where s(M(2n+1))less than or equal to 1 and is a CR invariant. The method that we adopt is analysis of the CR invariant Laplacian. We also explain the geometric origin of this question. (C) 1996 Academic Press, Inc.