Critical exponents of normal subgroups in higher rank

Abstract

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 ${\Gamma }^{\prime }◃\Gamma$ is a Zariski dense normal subgroup, then the limit cones of $\Gamma$ and ${\Gamma }^{\prime }$ 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 ${\delta }_{\varphi }({\Gamma }^{\prime })\ge \frac{1}{2}{\delta }_{\varphi }(\Gamma )$. Eventually, we show that if ${\Gamma }^{\prime }\backslash \Gamma$ is amenable, these critical exponents coincide.