Weighted cogrowth formula for free groups (Q784885)

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