We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group Cay(Fn) by an arbitrary subgroup G of Fn. 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\Cay(Fn) to the Poincaré exponent of G. Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.
20E08, 20F65 (Primary), 60J15, 60B15 (Secondary)
|Publication status||Published - 2018 Feb 22|
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