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.