The lengths of conjugators in the model filiform groups

TL;DR

This work determines the asymptotic growth of the conjugator length function for discrete model filiform groups $\Gamma_d = \mathbb{Z}^d \rtimes_{\phi_d} \mathbb{Z}$, showing ${\rm{CL}}_{\Gamma_d}(n) \simeq n^d$. The authors develop a novel inductive framework built on the center-to-quotient structure $\Gamma_d / \langle a_d \rangle \cong \Gamma_{d-1}$, and they combine precise control of centralisers, bounds on p-th roots, and Bézout-type arithmetic lemmas to bound conjugators. Key technical ingredients include a detailed normal form and distance analysis, a complete description of centralisers in $\Gamma_d$, quantitative root-length bounds, and a constructive lifting argument that leverages the Dehn function of $\Gamma_{d-1}$. The results extend the landscape of conjugator length phenomena beyond previously known cases, illustrating that polynomial conjugator lengths of arbitrary degree occur in finitely presented groups and highlighting the intricate geometry of conjugacy in nilpotent-like groups.

Abstract

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 $d$.

Theorems & Definitions (28)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2: BridsonPittet
• Lemma 2.3
• Theorem 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• Lemma 3.4