# 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 $\Gamma' \triangleleft \Gamma$ 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.
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Cited literature [16 references]

https://hal.archives-ouvertes.fr/hal-02862820
Contributor : Samuel Tapie <>
Submitted on : Tuesday, June 9, 2020 - 4:54:28 PM
Last modification on : Thursday, June 11, 2020 - 4:29:21 AM

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exposant critique.pdf
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### Identifiers

• HAL Id : hal-02862820, version 1
• ARXIV : 2006.05730

### Citation

Olivier Glorieux, Samuel Tapie. Critical exponents of normal subgroups in higher rank. 2020. ⟨hal-02862820⟩

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