Groups with fast-growing conjugator length functions

We construct the first examples of finitely presented groups whose conjugator length function is exponential; these are central extensions of groups of the form F m ⋊ F 2 Fₘ F₂ upper F Subscript m Baseline right normal factor semidirect product upper F 2 . Further, we use a fiber product construction to exhibit a family of finitely presented groups Γ k Γ ₖ normal upper Gamma Subscript k , where for each k , the conjugator length function of Γ k Γ ₖ normal upper Gamma Subscript k grows like functions lying in the k -th level of the Grzegorczyk hierarchy of primitive recursive functions.