We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group ${\rm Cay}(F_n)$ by an arbitrary subgroup $G$ of $F_n$. Our main result, which generalizes Grigorchuk's cogrowth formula to variable edge lengths, provides a formula relating the bottom of the spectrum of weighted Laplacian on $G \backslash {\rm Cay}(F_n)$ to the Poincaré exponent of $G$. Our main tool is the Patterson–Sullivan theory for Cayley graphs with variable edge lengths.

中文翻译：

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自由群体的加权共生公式

我们研究自由组$ {\ rm Cay}（F_n）$的任意子组$ G $ $ F_n $的Cayley图商的几何，分析和概率指标之间的关系。我们的主要结果将Grigorchuk的共生公式推广为可变的边长，提供了一个公式，该公式将$ G \反斜杠{\ rm Cay}（F_n）$上加权拉普拉斯算子的频谱底部与$ G $的Poincaré指数相关。我们的主要工具是边缘长度可变的Cayley图的Patterson–Sullivan理论。