Abstract
Full residual finiteness growth of a finitely generated group G measures how efficiently word metric n-balls of G inject into finite quotients of G. We initiate a study of this growth over the class of nilpotent groups. When the last term of the lower central series of G has finite index in the center of G we show that the growth is precisely n b, where b is the product of the nilpotency class and dimension of G. In the general case, we give a method for finding an upper bound of the form n b where b is a natural number determined by what we call a terraced filtration of G. Finally, we characterize nilpotent groups for which the word growth and full residual finiteness growth coincide.
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K. B. supported in part by NSF grant DMS-1405609.
D. S. supported in part by NSF grant DMS-1246989.
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Bou-Rabee, K., Studenmund, D. Full residual finiteness growths of nilpotent groups. Isr. J. Math. 214, 209–233 (2016). https://doi.org/10.1007/s11856-016-1358-x
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DOI: https://doi.org/10.1007/s11856-016-1358-x