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 Poincaré 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 $R\to +\infty$, $$N_{\Gamma }(R):=\#\left\{\gamma \in \Gamma ;d(o,\gamma \cdot o)\le R\right\}\sim C_{\Gamma }\frac{L(R)}{R^{\alpha }}{e}^{{\delta }_{\Gamma }R},$$ for some constant $C_{\Gamma }>0$.

20 pages