The lengths of conjugators in the model filiform groups

The conjugator length function of a finitely generated group $\mathrm{\Gamma }$ gives the optimal upper bound on the length of a shortest conjugator for any pair of conjugate elements in the ball of radius $n$ in the Cayley graph of $\mathrm{\Gamma }$. We prove that polynomials of arbitrary degree arise as conjugator length functions of finitely presented groups. To establish this, we analyse the geometry of conjugation in the discrete model filiform groups ${\mathrm{\Gamma }}_d={\mathbb{Z}}^d{⋊}_{\phi }\mathbb{Z}$ where is $\phi$ is the automorphism of ${\mathbb{Z}}^d$ that fixes the last element of a basis $a_1,\dots ,a_d$ and sends $a_i$ to $a_i{a}_{i+1}$ for $i<d$. The conjugator length function of ${\mathrm{\Gamma }}_d$ is polynomial of degree $d$.