Abstract
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.
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