The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as ${\mathrm{C}\mathrm{l}}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either ${\mathrm{C}\mathrm{l}}_G$ must be constant or ${\mathrm{C}\mathrm{l}}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.