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

Abstract

For every $\alpha \le \beta$ in a left neighborhood $[{\alpha }_0,1]$ of 1, a group $G(\alpha ,\beta )$ is constructed, the growth function of which satisfies $\mathrm{limsup}\frac{\log \log b_{G(\alpha ,\beta )}(r)}{\log r}=\alpha$ and $\mathrm{liminf}\frac{\log \log b_{G(\alpha ,\beta )}(r)}{\log r}=\beta$. 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.

arXiv commentFinal version to appear in Afrika Matematika