A Finitary Version of Gromov’s Polynomial Growth Theorem

Yehuda Shalom, Terence Tao

Type: Article

Publication Date: 2010-10-15

Citations: 88

DOI: https://doi.org/10.1007/s00039-010-0096-1

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.

Locations

Geometric and Functional Analysis - View - PDF
Geometric and Functional Analysis - View - PDF

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