A Finitary Version of Gromov’s Polynomial Growth Theorem
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A Finitary Version of Gromov’s Polynomial Growth Theorem

  • Author(s): Shalom, Yehuda
  • Tao, Terence
  • et al.
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 some R 0 > exp(exp(Cd C )) for which the number of elements in a ball of radius R 0 in a Cayley graph of G is bounded by $${R_0^d}$$ , then 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.

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