Electronic Journal of Probability

Critical Constants for Recurrence on Groups of Polynomial Growth

David Revelle and Russ Thompson

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The critical constant for recurrence, $c_{rt}$, is an invariant of the quotient space $H/G$ of a finitely generated group. The constant is determined by the largest moment a probability measure on $G$ can have without the induced random walk on $H/G$ being recurrent. We present a description of which subgroups of groups of polynomial volume growth are recurrent. Using this we show that for such recurrent subgroups $c_{rt}$ corresponds to the relative growth rate of $H$ in $G$, and in particular $c_{rt}$ is either $0$, $1$ or $2$.

Article information

Electron. J. Probab., Volume 15 (2010), paper no. 23, 710-722.

Accepted: 16 April 2010
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

nilpotent group Schreier graph random walk recurrence volume growth

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Revelle, David; Thompson, Russ. Critical Constants for Recurrence on Groups of Polynomial Growth. Electron. J. Probab. 15 (2010), paper no. 23, 710--722. doi:10.1214/EJP.v15-773. https://projecteuclid.org/euclid.ejp/1464819808

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