ORBITAL COUNTING FOR SOME CONVERGENT GROUPS

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincare series converges at the critical exponent delta(Gamma). We obtain an explicit asymptotic for their orbital growth function. Namely, for any alpha is an element of]1, 2[ and any smooth slowly varying function L : R -> (0, + infinity), we construct N-dimensional Hadamard manifolds (X, g) of negative and pinched curvature, whose group of oriented isometrics possesses convergent geometrically finite subgroups Gamma such that, as R -> +infinity, N-Gamma(R) := #{gamma is an element of Gamma vertical bar d(o, gamma. o) <= R} similar to C-Gamma(o)L(R)/R-alpha e(delta Gamma R), for some Cr(o) > 0 depending on the base point o.