论文标题
在边界的晶格和双曲基团的熵光谱上
On the Lattice of Boundaries and the Entropy Spectrum of Hyperbolic Groups
论文作者
论文摘要
令$γ$为非元素双曲线群，$μ$为$γ$的概率。我们研究$ $ $ $ - 固定的动作，也称为边界动作，为$γ$。特别是，我们对$（γ，μ）$ - 边界的Furstenberg熵以及集合$ \ MATHCAL {BL}（γ，μ）边界的晶格理论和拓扑结构感兴趣。我们证明，所有双曲线群都具有无限的许多不同边界，这些边界达到了无限的一组不同的熵。此外，对于在非亚伯利亚自由组上的简单随机步行$ f_d $，我们确定有许多边界的熵大于$ \ frac {1} {2} {2}-ε$ $ times Poisson边界的熵，而等级$ d $很大。一路上获得了关于弗斯滕伯格熵的订单理论和连续性特性的独立兴趣的一般结果。这包括一个结果：在温和的假设下，边界熵的光谱$ \ MATHCAL {H} _ {\ text {bound}}}（γ，μ）$被关闭。
Let $Γ$ be a non-elementary hyperbolic group and $μ$ be a probability on $Γ$. We study the $μ$-proximal, stationary actions, also known as boundary actions, of $Γ$. In particular, we are interested in the spectrum of Furstenberg entropies of $(Γ,μ)$-boundaries, and the lattice-theoretic and topological structure of the set $\mathcal{BL}(Γ,μ)$ of boundaries. We prove that all hyperbolic groups have infinitely many distinct boundaries, which attain an infinite set of distinct entropies. Additionally, for simple random walks on non-abelian free groups $F_d$, we establish that there are infinitely many boundaries whose entropy is greater than $\frac{1}{2}-ε$ times the entropy of Poisson boundary, when the rank $d$ is large. General results of independent interest about the order-theoretic and continuity properties of Furstenberg entropy for countable groups are attained along the way. This includes the result that under mild assumptions, the spectrum of boundary entropies $\mathcal{H}_{\text{bound}}(Γ,μ)$ is closed.
