Heat kernel and counting problems for Kleinian groups

Given a group acting discretely and properly on a metric space, we are interested in estimating the number of points belonging to both the orbit of a given point and to a ball of fixed center when the radius of the ball goes to infinity. Any control of this asymptotic is a counting theorem.
In this talk we will focus on Kleinian groups, i.e groups acting on a hyperbolic space. We will start with the historical background, and proceed with the proposal of a strategy to obtain counting theorems based on the use of the heat kernel.