A Finitary Version of Gromov's Polynomial Growth Theorem

Yehuda Shalom; Terence Tao

doi:10.1007/s00039-010-0096-1

A Finitary Version of Gromov's Polynomial Growth Theorem

Yehuda Shalom^*, Terence Tao

^*Corresponding author for this work

University of California at Los Angeles

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

44 Scopus citations

Abstract

We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is someR₀ > exp(exp(Cd^C)) for which the number of elements in a ball of radius R₀ in a Cayley graph of G is bounded by R ^d₀then G has a finite-index subgroup which is nilpotent (of step < C^d). An effective bound on the finite index is provided if "nilpotent" is replaced by "polycyclic", thus yielding a non-trivial result for finite groups as well.

Original language	English
Pages (from-to)	1502-1547
Number of pages	46
Journal	Geometric and Functional Analysis
Volume	20
Issue number	6
DOIs	https://doi.org/10.1007/s00039-010-0096-1
State	Published - Dec 2010
Externally published	Yes

Funding

Funders	Funder number
Israel Science Foundation
National Science Foundation	500/05, 0701639, 0649473, DMS-0701639
John D. and Catherine T. MacArthur Foundation	DMS-0649473

Keywords

Polynomial growth
harmonic functions
nilpotent groups

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10.1007/s00039-010-0096-1

Cite this

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title = "A Finitary Version of Gromov's Polynomial Growth Theorem",

abstract = "We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is someR0 > exp(exp(CdC)) for which the number of elements in a ball of radius R0 in a Cayley graph of G is bounded by R d0then G has a finite-index subgroup which is nilpotent (of step < Cd). An effective bound on the finite index is provided if {"}nilpotent{"} is replaced by {"}polycyclic{"}, thus yielding a non-trivial result for finite groups as well.",

keywords = "Polynomial growth, harmonic functions, nilpotent groups",

author = "Yehuda Shalom and Terence Tao",

note = "Funding Information: 1.15 Acknowledgments. The authors thank Emmanuel Breuillard for valuable discussions, and the anonymous referee for corrections. The first author was supported by ISF and NSF grants number 500/05 and DMS-0701639 resp. The second author is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.",

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doi = "10.1007/s00039-010-0096-1",

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N1 - Funding Information: 1.15 Acknowledgments. The authors thank Emmanuel Breuillard for valuable discussions, and the anonymous referee for corrections. The first author was supported by ISF and NSF grants number 500/05 and DMS-0701639 resp. The second author is supported by a grant from the MacArthur Foundation, by NSF grant DMS-0649473, and by the NSF Waterman award.

PY - 2010/12

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N2 - We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is someR0 > exp(exp(CdC)) for which the number of elements in a ball of radius R0 in a Cayley graph of G is bounded by R d0then G has a finite-index subgroup which is nilpotent (of step < Cd). An effective bound on the finite index is provided if "nilpotent" is replaced by "polycyclic", thus yielding a non-trivial result for finite groups as well.

AB - We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is someR0 > exp(exp(CdC)) for which the number of elements in a ball of radius R0 in a Cayley graph of G is bounded by R d0then G has a finite-index subgroup which is nilpotent (of step < Cd). An effective bound on the finite index is provided if "nilpotent" is replaced by "polycyclic", thus yielding a non-trivial result for finite groups as well.

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KW - harmonic functions

KW - nilpotent groups

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A Finitary Version of Gromov's Polynomial Growth Theorem

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