The lengths of conjugators in the model filiform groups

The conjugator length function of a finitely generated group $Γ$ 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 $Γ$. 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 $Γ_d = \mathbb{Z}^d\rtimes_ϕ\mathbb{Z}$ where is $ϕ$ 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 $Γ_d$ is polynomial of degree $d$.

💡 Research Summary

The paper investigates the conjugator length function (CL) of finitely generated groups, focusing on the discrete model filiform groups Γₙ = ℤⁿ ⋊_ϕ ℤ, where the automorphism ϕ fixes the last basis element aₙ and sends aᵢ to aᵢ aᵢ₊₁ for i < n. The CL of a group G, denoted CL_G(n), is the smallest integer N such that any two conjugate elements represented by words of length ≤ n can be conjugated by a word of length ≤ N. While Dehn functions (measuring the difficulty of the word problem) are well understood, the class of functions that arise as CL has been largely unexplored; in particular, it was unknown whether arbitrary polynomial degrees could be realized.

The authors prove that for each integer d ≥ 1 the conjugator length function of Γ_d grows like a degree‑d polynomial: CL_{Γ_d}(n) ≍ n^d. This provides the first explicit family of finitely presented groups whose CL exhibits any prescribed polynomial degree, thereby answering the open question in the affirmative.

The proof consists of several key components:

1. Group Structure and Normal Forms.
   Γ_d admits the presentation
   \