JournalsggdVol. 14, No. 2pp. 349–368

Weighted cogrowth formula for free groups

  • Johannes Jaerisch

    Nagoya University, Japan
  • Katsuhiko Matsuzaki

    Waseda University, Tokyo, Japan
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Abstract

We investigate the relationship between geometric, analytic and probabilistic indices for quotients of the Cayley graph of the free group Cay(Fn){\rm Cay}(F_n) by an arbitrary subgroup GG of FnF_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\Cay(Fn)G \backslash {\rm Cay}(F_n) to the Poincaré exponent of GG. Our main tool is the Patterson–Sullivan theory for Cayley graphs with variable edge lengths.

Cite this article

Johannes Jaerisch, Katsuhiko Matsuzaki, Weighted cogrowth formula for free groups. Groups Geom. Dyn. 14 (2020), no. 2, pp. 349–368

DOI 10.4171/GGD/547