Patterson–Sullivan Measures and Growth of Relatively Hyperbolic Groups

We prove that for a relatively hyperbolic group,there is a sequence of relatively hyperbolic proper quotients such that their growth rates converge to the growth rate of the group.Under natural assumptions,a similar result holds for the critical exponent of a cusp-uniform action of the group on a hyperbolic metric space.As a corollary,we obtain that the critical exponent of a torsion-free geometrically finite Kleinian group can be arbitrarily approximated by those of proper quotient groups.This resolves a question of Dal’bo–Peigné–Picaud–Sambusetti.Our approach is based on the study of Patterson–Sullivan measures on Bowditch boundary of a relatively hyperbolic group and gives a series of results on growth functions of balls and cones.