Attainability of the minimal exponential growth rate for free products of finite cyclic groups

We consider free products of two finite cyclic groups of orders $2$ and $n$, where $n$ is a prime power. For any such group $\mathbb Z_2*\mathbb Z_n=\langle a,b\mid a^2=b^n=1\rangle$, we prove that the minimal growth rate $\alpha _n$ is attained on the set of generators $\{a,b\}$ and explicitly write out an integer polynomial whose maximal root is $\alpha_n$. In the cases of $n=3,4$, this result was obtained earlier by A. Mann. We also show that under sufficiently general conditions, the minimal growth rates of a group $G$ and of its central extension $\widetilde G$ coincide and that the attainability of one implies the attainability of the other. As a corollary, the attainability is proved for some cyclic extensions of the above-mentioned free products, in particular, for groups $\langle a,b\mid a^2=b^n\rangle$, which are groups of torus knots for odd $n$.