Abstract
In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius n centered at the identity. We prove that in the case of virtually polycyclic groups, this function is either exponential or polynomially bounded, and is polynomially bounded exactly when the group is virtually nilpotent. The proof is fairly short, and makes use of the fact that any polycyclic group has a subgroup of finite index which can be embedded as a lattice in a Lie group, as well as exponential radical of Lie groups and Dirichlet’s approximation theorem.
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References
Babenko I.K.: Closed geodesics, asymptotic volume, and characteristics of group growth. Izv. Akad. Nauk SSSR Ser. Mat. 52, 675–711 (1988)
Bourbaki N.: Lie Groups and Lie Algebras. Springer-Verlag, New York (1989)
Bass H.: The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. (3) 25, 603–614 (1972)
V. S. Guba and M. V. Sapir, On the Conjugacy Growth Function of Groups. \({{\tt arXiv:1003.1293v2}}\)
Hardy G.H., Wright E.M.: An Introduction to the Theory of Numbers. Oxford University Press Inc., New York (1979)
Osin D.V.: Exponential Radicals of Solvable Lie Groups. J. Algebra 248, 790–805 (2002)
Raghunathan M.S.: Discrete Subgroups of Lie Groups. Springer-Verlag, Berlin (1972)
Švarc A.S.: Volume invariants of coverings. Dokl. Akad. Nauk. 105, 32–34 (1955) (In Russian)
Wolf J.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Diff. Geom. 2, 421–446 (1968)
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Hull, M. Conjugacy growth in polycyclic groups. Arch. Math. 96, 131–134 (2011). https://doi.org/10.1007/s00013-011-0222-9
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DOI: https://doi.org/10.1007/s00013-011-0222-9