Archiv der Mathematik

, Volume 96, Issue 2, pp 131–134 | Cite as

Conjugacy growth in polycyclic groups



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.

Mathematics Subject Classification (2010)

20F69 20E45 20F16 


Conjugacy growth Polycyclic group 


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© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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