We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen–Margulis–Sullivan densities of geodesic planes, with degree given by the modified critical exponents.

We give a finitary criterion for the convergence of measures on non-elementary geometrically finite hyperbolic orbifolds to the unique measure of maximal entropy. We give an entropy criterion controlling escape of mass to the cusps of the orbifold. Using this criterion, we prove new results on the distribution of collections of closed geodesics on such an orbifold, and as a corollary, we prove the equidistribution of closed geodesics up to a certain length in amenable regular covers of geometrically finite orbifolds.

In this paper, we obtain several results on the commensurability of two Kleinian groups and their limit sets. We prove that two finitely generated subgroups G1 and G2 of an infinite co-volume Kleinian group G⊂Isom(H3) having Λ(G1)=Λ(G2) are commensurable. In particular, we prove that any finitely generated subgroup H of a Kleinian group G⊂Isom(H3) with Λ(H)=Λ(G) is of finite index if and only if H is not a virtually fibered subgroup.

In this paper we give the definition of a meromorphic function which is geometrically finite and investigate some properties of geometrically finite meromorphic functions and the Lebesgue measure of their Julia sets.