Geodesic
Attainability of the minimal exponential growth rate for free products of finite cyclic groups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 314-328.

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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 $n$. 
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     title = {Attainability of the minimal exponential growth rate for free products of finite cyclic groups},
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A. L. Talambutsa. Attainability of the minimal exponential growth rate for free products of finite cyclic groups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 314-328. http://geodesic.mathdoc.fr/item/TM_2011_274_a17/

[1] de lya Arp P., Busher M., “Svobodnye proizvedeniya s ob'edineniem i HNN-rasshireniya ravnomerno eksponentsialnogo rosta”, Mat. zametki, 67:6 (2000), 811–815 | DOI | MR

[2] Lindon R., Shupp P., Kombinatornaya teoriya grupp, Mir, M., 1980 | MR

[3] Talambutsa A.L., “Dostizhimost pokazatelya eksponentsialnogo rosta v svobodnykh proizvedeniyakh tsiklicheskikh grupp”, Mat. zametki, 78:4 (2005), 614–618 | DOI | MR | Zbl

[4] Talambutsa A.L., “Dostizhimost minimalnogo pokazatelya eksponentsialnogo rosta dlya nekotorykh fuksovykh grupp”, Mat. zametki, 88:1 (2010), 152–156 | DOI | MR | Zbl

[5] Talambutsa A.L., Dostizhimost minimalnogo pokazatelya rosta grupp s periodicheskimi sootnosheniyami, Dis. ... kand. fiz.-mat. nauk, Mat. in-t im. V.A. Steklova RAN, M., 2011 | Zbl

[6] Biberbakh L., Analiticheskoe prodolzhenie, Nauka, M., 1967 | MR

[7] de la Harpe P., Topics in geometric group theory, Univ. Chicago Press, Chicago, 2000 | MR | Zbl

[8] Conner G.R., “Central extensions of word hyperbolic groups satisfy a quadratic isoperimetric inequality”, Arch. Math., 65 (1995), 465–470 | DOI | MR | Zbl

[9] Mann A., “The growth of free products”, J. Algebra, 326:1 (2011), 208–217 | DOI | MR | Zbl

[10] Gromov M., Structures métriques pour les variét'es riemanniennes, Textes Math., 1, eds. J. Lafontaine, P. Pansu, Cedic/Fernand Nathan, Paris, 1981 | MR | Zbl

[11] Sambusetti A., “Growth tightness of free and amalgamated products”, Ann. Sci. École Norm. Supér. Sér. 4, 35 (2002), 477–488 | MR | Zbl

[12] Sambusetti A., “Minimal growth of non-Hopfian free products”, C. r. Acad. sci. Paris. Sér. 1: Math., 329 (1999), 943–946 | DOI | MR | Zbl

[13] Wilson J.S., “On exponential growth and uniformly exponential growth for groups”, Invent. math., 155:2 (2004), 287–303 | DOI | MR | Zbl