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.