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

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.