双曲群的统计极限律
Statistical limit laws for hyperbolic groups
摘要：
利用遍历论和符号动力学的技巧，我们得到了双曲群上实值函数的统计极限律。特别是，我们的结果适用于凸共紧群作用在$\text{CAT}(-1)$空间，并提供了精确的统计比较字长和位移。在将我们的方法推广到多维情形后，我们证明了阿贝尔化映射满足一个非退化的多维中心极限定理。我们还获得了群同态和位移函数的局部极限定理。
英文摘要：
Using techniques from ergodic theory and symbolic dynamics, we derive statistical limit laws for real valued functions on hyperbolic groups. In particular, our results apply to convex cocompact group actions on $\text{CAT}(-1)$ spaces, and provide a precise statistical comparison between word length and displacement. After generalising our methods to the multidimensional setting, we prove that the abelianisation map satisfies a non-degenerate multidimensional central limit theorem. We also obtain local limit theorems for group homomorphisms and for the displacement function associated to certain actions.
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