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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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Shalom, Y., Tao, T. A Finitary Version of Gromov’s Polynomial Growth Theorem. Geom. Funct. Anal. 20, 1502–1547 (2010). https://doi.org/10.1007/s00039-010-0096-1
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DOI: https://doi.org/10.1007/s00039-010-0096-1