A Finitary Version of Gromov's Polynomial Growth Theorem

Yehuda Shalom, Terence Tao

Research output: Contribution to journalArticlepeer-review


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


  • Polynomial growth
  • harmonic functions
  • nilpotent groups


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