Conjugacy growth of finitely generated groups

The conjugacy growth function of a finitely generated group measures the number of conjugacy classes in balls with respect to a word metric. We study the following natural question: Which functions can occur as the conjugacy growth function of finitely generated groups? Our main result answers the question completely. Namely we prove that a function f:N→Nf:N→N can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group if and only if ff is non-decreasing and bounded from above by anan for some a≥1a≥1. We also construct a finitely generated group GG and a subgroup H≤GH≤G of index 2 such that HH has only 2 conjugacy classes while the conjugacy growth of GG is exponential. In particular, conjugacy growth is not a quasi-isometry invariant.