Kleinian groups of divergence type

Abstract:

For a Kleinian group $\Gamma$ acting in the unit ball $B$ we consider the series ${\Sigma _{\gamma \in \Gamma }}{(1 - \left | {\gamma (0)} \right |)^2}$. If the series diverges, $\Gamma$ is said to be of divergence type. From the point of view of the ergodic properties of the group action it is essential to know whether or not $\Gamma$ is of divergence type. If $\Gamma$ is geometrically finite then $\Gamma$ is of divergence type if and only if it is of the first kind. However in the nongeometrically finite case it is not known whether there are any groups of divergence type. In this paper we give a geometric criterion which is sufficient to ensure divergence type and use this to construct an example of a nongeometrically finite Kleinian group of divergence type.