具有秩1等距的Hadamard空间乘积的渐近几何
Asymptotic Geometry in the product of Hadamard spaces with rank one isometries
摘要：
本文研究了局部紧Hadamard空间的乘积$\XX=\XX_1\times \XX_2$上的离散群$Γ$通过等距作用的渐近性质。动机来自于这样一个事实，即有限域上的Kac-Moody群，可以看作是函数域上算术群的推广，属于这类群。因此，人们可能会问，在[MR 1437472]和[MR 1933790]中的高阶李群的离散子群的经典性质是否在这种情况下成立。本文的第一部分描述了$Γ$的几何极限集的结构，并证明了类似于Benoist在[MR 1437472]中的结果。第二部分研究了$\XX$中轨道点的指数增长性率$δ_θ(Γ)$，并给出了一个所谓的“斜率”$θ\in (0,π/2)$，它适当地推广了高阶临界指数。类似于Quint在[MR 1933790]中的结果，我们证明了作为$θ$的函数的$δ_θ(Γ)$的齐次扩张$Ψ_Γ$到$\RR_{\ge 0}^2$是上半连续的和凹的。
英文摘要：
In this article we study asymptotic properties of certain discrete groups $Γ$ acting by isometries on a product $\XX=\XX_1\times \XX_2$ of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $Γ$ and prove statements analogous to the results of Benoist in [MR1437472]. The second part is concerned with the exponential growth rate $δ_θ(Γ)$ of orbit points in $\XX$ with a prescribed so-called "slope" $θ\in (0,π/2)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint's result in [MR1933790] we show that the homogeneous extension $Ψ_Γ$ to $\RR_{\ge 0}^2$ of $δ_θ(Γ)$ as a function of $θ$ is upper semi-continuous and concave.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
