Weighted cogrowth formula for free groups

Johannes Jaerisch; Katsuhiko Matsuzaki

Weighted cogrowth formula for free groups

School of Education

Research output: Contribution to journal › Article › peer-review

Abstract

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)

Original language	English
Journal	Unknown Journal
Publication status	Published - 2018 Feb 22

ASJC Scopus subject areas

General

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title = "Weighted cogrowth formula for free groups",

abstract = "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 Poincar{\'e} exponent of G. Our main tool is the Patterson-Sullivan theory for Cayley graphs with variable edge lengths.20E08, 20F65 (Primary), 60J15, 60B15 (Secondary)",

author = "Johannes Jaerisch and Katsuhiko Matsuzaki",

year = "2018",

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journal = "Nuclear Physics A",

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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 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)

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 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)

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