Groups with fast-growing conjugator length functions

We construct the first examples of finitely presented groups where the conjugator length function is exponential; these are central extensions of groups of the form $F_m \rtimes F_2$. Further, we use a fibre product construction to exhibit a family of finitely presented groups $\Gamma_k$ where, for each $k$, the conjugator length function of $\Gamma_k$ grows like functions in the $k$-th level of the Grzegorczyk hierarchy of primitive recursive functions.