HAUSDORFF DIMENSION OF DIRECTIONAL LIMIT SETS FOR SELF-JOININGS OF HYPERBOLIC MANIFOLDS

The classical result of Patterson and Sullivan says that for a nonelementary convex cocompact subgroup Gamma < SO degrees(n, 1), n >= 2, the Hausdorff dimension of the limit set of Gamma is equal to the critical exponent of Gamma. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let Delta be a finitely generated group and rho(i) :Delta -> SO degrees (n(i),1) be a convex cocompact faithful representation of Delta for 1 <= i <= k. Associated to rho= (rho 1,...,rho k), we consider the following self-joining subgroup of Pi(k)(i =1) SO(n(i), 1): Gamma= (Pi(k)(i=1) rho(i)) (Delta) = {(rho(1) (g,..., rho(k) (g)): g is an element of Delta)}. 1. Denoting by Lambda subset of Pi S-k(i=1)i(n)-1 the limit set of Gamma, we first prove that dim(H Lambda)= max (1 <= i <= k) delta rho(i), where d.i is the critical exponent of the subgroup rho(i) (Delta). 2. Denoting by Lambda(u) subset of Lambda the u-directional limit set for each u = (u(1),..., u(k)) in the interior of the limit cone of Gamma, we obtain that for k <= 3, psi Gamma(u)/max(i) u(i) <= dim(H)Lambda(u) <= Psi Gamma(u) /min(i) u(i), where psi(Gamma) : R-k -> R boolean OR {-infinity} is the growth indicator function of Gamma.