论文标题
较高等级的正常亚组的关键指数
Critical exponents of normal subgroups in higher rank
论文作者
论文摘要
我们研究了更高排名半简单的真实线性谎言组$ g $的离散子组的关键指数。让我们修复$ g $的Lie代数的Cartan子空间$ \ Mathfrak A \ subset \ Mathfrak g $。我们表明，如果$γ<g $是一个离散的组，而$γ'\ triangleleftγ$是zariski密集的正常子组，那么$γ$的极限锥和$γ'$ in $ \ mathfrak a $ comincide。此外，对于所有线性形式$ ϕ：\ mathfrak a \ to \ mathbb r $在此极限锥上呈阳性，$ ϕ $的关键指数满足$ \displayStyletyleΔ__ϕ（γ'\ geq'）\ geq'\ geq \ frac 1 2δ__ϕ（γ）$。最终，我们表明，如果$γ'\backslashγ$适合，这些关键指数一致。
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 $Γ' \triangleleft Γ$ is a Zariski dense normal subgroup, then the limit cones of $Γ$ and $Γ'$ 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 $\displaystyle δ_ϕ(Γ') \geq \frac 1 2 δ_ϕ(Γ)$. Eventually, we show that if $Γ'\backslash Γ$ is amenable, these critical exponents coincide.
