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

A. L. Talambutsa

Informatics and Automation, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 314-328

Résumé

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 $n$.