Counting for some convergent groups

Abstract

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincar\'e series converges at the critical exponent $\delta_\Gamma$. We obtain an explicit asymptotic for their orbital growth function. Namely, for any $\alpha \in ]1, 2[ $ and any slowly varying function $L : \mathbb R\to (0, +\infty)$, we construct $N$-dimensional Hadamard manifolds $(X, g)$ of negative and pinched curvature, whose group of oriented isometries admits convergent geometrically finite subgroups $\Gamma$ such that, as $R\to +\infty$, $$ N_\Gamma(R):= \#\left\{\gamma\in \Gamma \; ; \; d(o, \gamma \cdot o)\leq R\right\} \sim C_\Gamma \frac{L(R)}{R^\alpha} \ e^{\delta_\Gamma R}, $$ for some constant $C_\Gamma >0$.