A Lower Bound for the Exponent of Convergence of Normal Subgroups of Kleinian Groups

We give a short new proof that for each non-elementary Kleinian group $\mathrm{\Gamma }$, the exponent of convergence of an arbitrary non-trivial normal subgroup is bounded below by half of the exponent of convergence of $\mathrm{\Gamma }$, and that strict inequality holds if $\mathrm{\Gamma }$ is of divergence type.