Counting problems in infinite measure.

Given a group Γ acting properly discontinuously and by isometries on a metric space X, one can wonder how grows the orbit of a given point. More precisely, given two points x, y ∈ X and ρ > 0, we define the orbital function as NΓ(x, y, ρ) := #(Γ · y ∩ B(x, ρ)) , where B(x, ρ) denotes the ball centred at x of radius ρ. A counting problem con- sists to estimate the orbital function when ρ → ∞. In the setting of groups acting on hyperbolic spaces this question was widely in- vestigated for decades, with mainly two different approaches: an analytical one relying on Selberg’s pre-trace formula, due to Huber in the 50’s, and a dynami- cal one relying on the mixing of the geodesic flow, due to Margulis in the late 60’s. During the talk, we shall describe Margulis’ dynamical method in order to mo- tivate the introduction of the Brownian motion. Combined with the use of the pre-trace formula, we shall establish a counting theorem linking the heat kernel of the quotient manifold and the orbital function. If the time allows it, we also shall review a couple of corollaries of the approach.