Growth behaviors in the range $e^{r^\alpha}$

TL;DR

This paper constructs groups with precisely controlled oscillating growth functions, revealing new invariants and demonstrating the complexity of growth behaviors in finitely generated groups.

Contribution

It introduces a method to explicitly construct groups with prescribed oscillating growth rates and identifies new invariants related to oscillation frequency.

Findings

Constructed groups with growth functions oscillating between specified bounds.

Established existence of uncountably many groups with comparable growth behaviors.

Identified new invariants linked to oscillation period and frequency.

Abstract

For every $\alpha \leq \beta$ in a left neighborhood $[\alpha_0,1]$ of 1, a group $G(\alpha,\beta)$ is constructed, the growth function of which satisfies $\limsup \frac{\log \log b_{G(\alpha,\beta)}(r)}{\log r}=\alpha$ and $\liminf \frac{\log \log b_{G(\alpha,\beta)}(r)}{\log r}=\beta$. When $\alpha=\beta$, this provides an explicit uncountable collection of groups with growth functions strictly comparable. On the other hand, oscillation in the case $\alpha < \beta$ explains the existence of groups with non comparable growth functions. Some period exponents associated to the frequency of oscillation provide new group invariants.