Abstract
We consider free products of two finite cyclic groups of orders 2 and n, where n is a prime power. For any such group ℤ2 * ℤ n = 〈a, b | a 2 = b n = 1〉, we prove that the minimal growth rate α n is attained on the set of generators {a, b} and explicitly write out an integer polynomial whose maximal root is α 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 \(\tilde G\) 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 〈a, b | a 2 = b n〉, which are groups of torus knots for odd n.
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Original Russian Text © A.L. Talambutsa, 2011, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Vol. 274, pp. 314–328.
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Talambutsa, A.L. Attainability of the minimal exponential growth rate for free products of finite cyclic groups. Proc. Steklov Inst. Math. 274, 289–302 (2011). https://doi.org/10.1134/S0081543811060186
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DOI: https://doi.org/10.1134/S0081543811060186