A Finitary Version of Gromov's Polynomial Growth Theorem

Yehuda Shalom*, Terence Tao

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

43 Scopus citations


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.

Original languageEnglish
Pages (from-to)1502-1547
Number of pages46
JournalGeometric and Functional Analysis
Issue number6
StatePublished - Dec 2010
Externally publishedYes


FundersFunder number
National Science Foundation500/05, DMS-0701639
Directorate for Mathematical and Physical Sciences0701639, 0649473
John D. and Catherine T. MacArthur FoundationDMS-0649473
Israel Science Foundation


    • Polynomial growth
    • harmonic functions
    • nilpotent groups


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