Is there a finitely **presented** group with exactly linear conjugacy growth which is not virtually cyclic?


Here, conjugacy growth function $c(n)$ counts the number of conjugacy classes in the $n$-ball, and it is exactly linear if $c(n) \sim An$ for some constant $A \in (0, \infty)$.


Note that by the proof of Corollary 2.5 in von Puttkamer-Wu's paper: Linear groups, conjugacy growth, and classifying spaces for families of subgroups (IMRN, 2019), this group has to be non-hyperbolic.



Linear conjugacy growth function
The conjugacy growth function $f(n)$ is the number of conjugacy classes intersecting a ball of radius $n$. So if you take Ivanov's or Osin's examples of finitely generated groups with finite number of conjugacy classes, the groups are not virtually cyclic, but the conjugacy growth function is constant. If you take the modification of Ivanov'a example $G=\langle a,b \rangle$ where $a$ is of infinite order and every element of $G$ is a conjugate of a power of $a$, then the growth function of that group is going to be linear. For references see [this paper](https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-54/issue-1/On-the-conjugacy-growth-functions-of-groups/10.1215/ijm/1299679750.full?tab=ArticleFirstPage).


**Update** These examples are not finitely presented. There are no known finitely presented examples.


