Hausdorff Dimension of Limit Sets for Spherical CR Manifolds

LetM2n+1(n⩾1) be a compact, spherical CR manifold. SupposeM2n+1is its universal cover andΦ : M2n+1→S2n+1is on injective CR developing map, whereS2n+1is the standard unit sphere in the complex (n+1)-spaceCn+1, thenM2n+1is of the quotient formΩ/Λ, whereΩis a simply connected open set inS2n+1, andΛis a complex Klein group acting onΩproperly discontinuously. In this paper, we show that if the CR Yamabe invariant ofM2n+1is positive, then the Carnot Hausdorff dimension of the limit set ofΛis bounded above byn·s(M2n+1), wheres(M2n+1)⩽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.