Weighted cogrowth formula for free groups

Abstract

We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group $\mathrm{Cay}(F_n)$ endowed with variable edge lengths, 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 $G\backslash \mathrm{Cay}(F_n)$ to the Poincaré exponent of $G$. Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.