Asymptotic growth and least common multiples in groups

TL;DR

This paper explores the relationship between growth functions in groups and residual finiteness, establishing criteria for nilpotency and resolving a question about free groups' growth bounds.

Contribution

It introduces new connections between word/subgroup growth and residual finiteness, and proves a key lower bound for free groups' growth functions.

Findings

Nilpotency characterized by asymptotic growth behavior

Resolved Bogopolski's question on free groups

Established new lower bounds for growth functions

Abstract

In this article we relate word and 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 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 arena resolves a question of Bogopolski from the Kourovka notebook concerning lower bounds of one of these functions for nonabelian free groups.