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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 $Γ< G$ is a discrete group, and $Γ&#39; \triangleleft Γ$ is a Zariski dense normal subgroup, then the limit cones of $Γ$ and $Γ&#39;$ in $\mathfrak a$ coincide. Moreover, for all linear form $ϕ: \mathfrak a\to \mathbb R$ positive on this limit cone, the critical exponents in the direction of $ϕ$ satisfy $\displaystyle δ_ϕ(Γ&#39;) \geq \frac 1 2 δ_ϕ(Γ)$. Eventually, we show that if $Γ&#39;\backslash Γ$ is amenable, these critical exponents coincide.