Growth behaviors in the range $e^{r^α}$

For every $α\leq β$ in a left neighborhood $[α_0,1]$ of 1, a group $G(α,β)$ is constructed, the growth function of which satisfies $\limsup \frac{\log \log b_{G(α,β)}(r)}{\log r}=α$ and $\liminf \frac{\log \log b_{G(α,β)}(r)}{\log r}=β$. When $α=β$, this provides an explicit uncountable collection of groups with growth functions strictly comparable. On the other hand, oscillation in the case $α< β$ explains the existence of groups with non comparable growth functions. Some period exponents associated to the frequency of oscillation provide new group invariants.

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