The lengths of conjugators in the model filiform groups

The conjugator length function of a finitely generated group $\Gamma$ 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 $\Gamma$. 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 $\Gamma_d = \mathbb{Z}^d\rtimes_\phi\mathbb{Z}$ 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 $a_ia_{i+1}$ for $i<d$. The conjugator length function of $\Gamma_d$ is polynomial of degree $d$.