Critical exponents of normal subgroups in higher rank

We study the critical exponents of discrete subgroups of a higher rank semi-simple real linear Lie group $G$. Let us fix a Cartan subspace $\mathfrak a\subset \mathfrak g$ of the Lie algebra of $G$. We show that if $\Gamma< G$ is a discrete group, and $\Gamma' \triangleleft \Gamma$ is a Zariski dense normal subgroup, then the limit cones of $\Gamma$ and $\Gamma'$ in $\mathfrak a$ coincide. Moreover, for all linear form $\phi : \mathfrak a\to \mathbb R$ positive on this limit cone, the critical exponents in the direction of $\phi$ satisfy $\displaystyle \delta_\phi(\Gamma') \geq \frac 1 2 \delta_\phi(\Gamma)$. Eventually, we show that if $\Gamma'\backslash \Gamma$ is amenable, these critical exponents coincide.