Asymptotic growth and least common multiples in groups

Abstract

In this article, we relate word and normal subgroup growth to certain functions that arise in the quantification of residual finiteness. One consequence of this endeavor is a pair of results that equate the virtual nilpotency of a finitely generated group with the asymptotic behavior of these functions. The second half of this article investigates the asymptotic behavior of two of these functions. Our main result in this area resolves a question of Bogopolski from the Kourovka notebook concerning lower bounds of one of these functions for non-abelian free groups.