Abstract
Consider the Poincaré series of order t for a discrete Moebius group acting on the n-dimensional upper half-space. If the point at infinity is a horocyclic limit point or a Garnett point, then the series diverges for any positive number t. If the point at infinity is an ordinary point or a cusped parabolic fixed point, then the series converges for any t which is greater than n-1. If the point at infinity is an atom for the Patterson-Sullivan measure, then the series converges for any t which is equal to or greater than the critical exponent of the group.
