论文标题
计数和边界限制gromov-Hyperbolic群体表示的定理
Counting and boundary limit theorems for representations of Gromov-hyperbolic groups
论文作者
论文摘要
考虑到具有有限的对称生成集的Gromov-Hyperbolic Group $ g $，我们研究了$ g $的线性表示，在相关Cayley图的球体上进行计数测量的统计数据。更普遍地，我们获得了较大的亚胶功能的弱定律，与经典的fekete引理相呼应。对于强烈的不可还原和近端表示，我们证明了具有浆果的计数中心限制定理 - 类型错误率和指数较大的较大偏差估计值。此外，在相同的环境中，我们显示了沿着测地射线与布朗运动的插值归一化矩阵规范的收敛性，以及迭代对数的功能定律，与随机矩阵产物理论中的类似结果相似。我们计算的大偏差估计值对Kaimanovich-Kapovich-Schupp的问题提供了积极的答案。在大多数情况下，我们的计数限制定理将从帕特森（Patterson）的较强限制定律中获得 - 苏利文（Sullivan）对组边界的措施。
Given a Gromov-hyperbolic group $G$ endowed with a finite symmetric generating set, we study the statistics of counting measures on the spheres of the associated Cayley graph under linear representations of $G$. More generally, we obtain a weak law of large numbers for subadditive functions, echoing the classical Fekete lemma. For strongly irreducible and proximal representations, we prove a counting central limit theorem with a Berry--Esseen type error rate and exponential large deviation estimates. Moreover, in the same setting, we show convergence of interpolated normalized matrix norms along geodesic rays to Brownian motion and a functional law of iterated logarithm, paralleling the analogous results in the theory of random matrix products. Our counting large deviation estimates provide a positive answer to a question of Kaimanovich--Kapovich--Schupp. In most cases, our counting limit theorems will be obtained from stronger almost sure limit laws for Patterson--Sullivan measures on the boundary of the group.
