# upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups

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)?

• 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.
– YCor
Commented Oct 3, 2023 at 14:02
• 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. Commented Oct 3, 2023 at 14:30
• 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.
– YCor
Commented Oct 3, 2023 at 17:13