To every finitely generated group, one can assign the conjugacy growth function that counts the number of conjugacy classes intersecting a ball of radius $n$. Results of Ivanov and Osin show that the conjugacy growth function may be constant even if the (ordinary) growth function is exponential. The aim of this paper is to provide conjectures, examples and statements that show that in “normal” cases, groups with exponential growth functions also have exponential conjugacy growth functions.
"On the conjugacy growth functions of groups." Illinois J. Math. 54 (1) 301 - 313, Spring 2010. https://doi.org/10.1215/ijm/1299679750