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Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of radius $n$ in the Cayley graph of $G$ with respect to $S$.

Suppose $G$ is a polycyclic group that is not virtually nilpotent, then $G$ has an exponential conjugacy growth function (https://arxiv.org/pdf/1006.1064.pdf).

I was wondering if anybody has computed an upper bound of the conjugacy exponential rate (for a given generating set)?

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  • $\begingroup$ What do you mean by "explicitly"? the exponential growth is intrinsic to the group, but (like ordinary growth) the exponential rate depends on a choice of generating subset. In other words, this is not a number, but a function from the set of finite generating subsets, to the set of positive numbers. $\endgroup$
    – YCor
    Commented Oct 3, 2023 at 14:02
  • $\begingroup$ Thanks for pointing that out. I saw some papers that prove certain non-virtually nilpotent groups have exponential conjugacy growth functions with respect to certain generating sets. I was trying to find out that if anybody has computed an upper bound for the rate. $\endgroup$
    – ghc1997
    Commented Oct 3, 2023 at 14:30
  • $\begingroup$ This is a bit vague. Yes, of course. The exponential growth rate (w.r.t. the given generating subset) is one. Also, the conjugacy growth of the free group with as many generators is one — I'm not sure what are the known estimates. The question seems a bit open-ended. $\endgroup$
    – YCor
    Commented Oct 3, 2023 at 17:13

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