The lengths of conjugators in the model filiform groups

TL;DR

This paper investigates the conjugator length function in certain finitely generated groups, showing that polynomial functions of arbitrary degree can serve as these bounds, with a detailed analysis of model filiform groups' geometry.

Contribution

It demonstrates that conjugator length functions can be polynomial of any degree, providing explicit examples within the class of model filiform groups.

Findings

Conjugator length functions can be polynomial of any degree.

The geometry of conjugation in model filiform groups is explicitly analyzed.

Polynomial degree of conjugator length functions matches the group's dimension.

Abstract

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 $d$.