TY - JOUR
T1 - Weighted cogrowth formula for free groups
AU - Jaerisch, Johannes
AU - Matsuzaki, Katsuhiko
N1 - Funding Information:
Acknowledgements. The authors would like to thank the anonymous referee for his valuable comments which helped to improve the presentation of the paper. This work was supported by JSPS KAKENHI 16K13767.
Publisher Copyright:
© European Mathematical Society.
PY - 2020
Y1 - 2020
N2 - 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 Gn Cay.Fn/ to the Poincaré exponent of G. Our main tool is the Patterson–Sullivan theory for metric trees.
AB - 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 Gn Cay.Fn/ to the Poincaré exponent of G. Our main tool is the Patterson–Sullivan theory for metric trees.
KW - Bottom of spectrum
KW - Cayley graph of free group
KW - Cogrowth
KW - Discrete Laplacian
KW - Green function
KW - Patterson–Sullivan theory
KW - Spectral radius
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U2 - 10.4171/GGD/547
DO - 10.4171/GGD/547
M3 - Article
AN - SCOPUS:85091287715
SN - 1661-7207
VL - 14
SP - 349
EP - 368
JO - Groups, Geometry, and Dynamics
JF - Groups, Geometry, and Dynamics
IS - 2
ER -