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We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group by an arbitrary subgroup of . 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 to the Poincaré exponent of . Our main tool is the Patterson–Sullivan theory for Cayley graphs with variable edge lengths.
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Johannes Jaerisch, Katsuhiko Matsuzaki, Weighted cogrowth formula for free groups. Groups Geom. Dyn. 14 (2020), no. 2, pp. 349–368DOI 10.4171/GGD/547