Vol. 17 No. 8 (2025): On Growth of Free Burnside Groups

Published: 2025-08-20

Articles

  • Articles

    On Growth of Free Burnside Groups

    Arman Bayramyan
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    Abstract

    In his seminal monograph, S. I. Adian established a lower bound on the growth of the free Burnside group $B(2,n)$ for odd $n\geq 665$. Building on Adian's methods, we extend this result to free Burnside groups $B(m,n)$ of arbitrary rank $m\geq 2$. As a consequence, we obtain growth estimates for a broad class of finitely generated groups.

    References

    S.I. Adian, The Burnside Problem and Identities in Groups, Springer-Verlag, 1979.

    S.I. Adian and V.S. Atabekyan, Characteristic properties and uniform non-amenability of n-periodic products of groups. Izv. Math., 79 (2015), no. 6, pp. 1097-1110.

    S.I. Adian and V.S. Atabekyan, n-torsion groups. J. Contemp. Math. Anal., 54 (2019), no. 6, pp. 319-327.

    V.S. Atabekyan, Monomorphisms of free Burnside groups. Math. Notes, 86 (2009), no. 4, pp. 457-462.

    V.S. Atabekyan, Uniform non-amenability of subgroups of free Burnside groups of odd period. Math. Notes, 85 (2009), no. 4, pp. 496-502.

    V.S. Atabekyan, H.T. Aslanyan and S.T. Aslanyan, Powers of subsets in free periodic groups. Proceedings of the YSU A: Physical and Mathematical Sciences, 56 (2022), no. 2, pp. 43-48.

    V.S. Atabekyan and A.A. Bayramyan, Probabilistic identities in $n$-torsion groups. J. Contemp. Math. Anal., 59 (2024), no. 6, pp. 455-459.

    V.S. Atabekyan and V.H. Mikaelian, On the product of subsets in periodic groups. J. Contemp. Math. Anal., 57 (2022), no. 6, pp. 395-398.

    R. Coulon, Growth of periodic quotients of hyperbolic groups. Algebr. Geom. Topol., 13 (2013), no. 6, pp. 3111-3133.

    R.I. Grigorchuk, Degrees of growth of finitely generated groups, and the theory of invariant means. Math. USSR Izv., 25 (1985), no. 2, pp. 259-300.

    M. Gromov, Groups of polynomial growth and expanding maps. Publications Mathématiques de L'Institut des Hautes Scientifiques, 53 (1981), pp. 53-73.

    J. Milnor, Growth of finitely generated solvable groups. J. Differential Geom., 2 (1968), no. 4, pp. 447-449.

    D.V. Osin, Uniform non-amenability of free Burnside groups. Arch. Math. (Basel), 88 (2007), no. 5, pp. 403-412.

    J. Tits, Free subgroups in linear groups. J. Algebra, 20 (1972), no. 2, pp. 250-270.

    J.A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Differential Geom., 2 (1968), no. 4, pp. 421-446.