Counting for some convergent groups

TL;DR

This paper constructs examples of negatively curved manifolds with specific geometric and dynamical properties, including non-ergodic geodesic flows and convergent Poincaré series, providing explicit asymptotics for orbital growth functions.

Contribution

It introduces new examples of geometrically finite manifolds with convergent Poincaré series and detailed asymptotic behavior of their orbital growth, expanding understanding of negative curvature dynamics.

Findings

Existence of manifolds with non-ergodic geodesic flow and convergent Poincaré series.

Explicit asymptotic formulas for orbital growth functions.

Construction of manifolds with prescribed growth rates involving slowly varying functions.

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 $C_\Gamma >0$.