The convergence of the Poincaré series on the limit set of a discrete group in several dimensions (Q1092280)

Let \(\Gamma\) be a discrete subgroup of the Möbius group acting on the ball in \(R^ n\). The author looks at the convergence of Poincaré series on L; the limit set of \(\Gamma\). He notes that one can extend the definition of Hedlund points, horocyclic limit points and Garnett points to this case. The results on these points are very precise generalizations of the results of Nicholls, Pommerenke and Sullivan for the case of the unit disk. The key to his results as in the one variable case is how many images of the origin lie in a region at \(\xi\), typically a cone or horosphere. His measurement of this accumulation index at \(\xi\in L\) is given by: \[ k(\xi)=\lim_{\sigma \in \Gamma}(\log | J(\sigma (\xi))|)/(\log | J(\sigma (0))|), \] where J is the Jacobian of the matrix \(\sigma\).