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 $\mathfrak a\subset \mathfrak g$oftheLiealgebraof$G$\mathrm{.}Weshowthatif$\Gamma< G$ is a discrete group, and ${\mathrm{\Gamma }}^{\mathrm{\prime }}◃\mathrm{\Gamma }$ is a Zariski dense normal subgroup, then the limit cones of $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ in $\mathfrak{a}$ coincide. Moreover, for all linear form $\varphi :\mathfrak{a}\to \mathbb{R}$ positive on this limit cone, the critical exponents in the direction of $\varphi$ satisfy $\displaystyle \delta_\phi(\Gamma’) \geq \frac 1 2 \delta_\phi(\Gamma)$\mathrm{.}Eventually,weshowthatif$\Gamma’\backslash \Gamma$ is amenable, these critical exponents coincide.