Linear Diophantine equations and conjugator length in 2-step nilpotent groups

TL;DR

This paper investigates bounds on conjugator lengths in 2-step nilpotent groups using solutions to linear Diophantine equations, revealing cases where these bounds are optimal and demonstrating polynomial growth of conjugator length functions.

Contribution

It introduces new upper bounds on conjugator lengths in 2-step nilpotent groups and constructs examples with polynomial growth of conjugator length functions.

Findings

Bounds are sharp in some cases

Conjugator length can grow polynomially with degree m+1

Small integral solutions are key to establishing bounds

Abstract

We establish upper bounds on the lengths of minimal conjugators in 2-step nilpotent groups. These bounds exploit the existence of small integral solutions to systems of linear Diophantine equations. We prove that in some cases these bounds are sharp. This enables us to construct a family of finitely generated 2-step nilpotent groups $(G_m)_{m\in\mathbb{N}}$ such that the conjugator length function of $G_m$ grows like a polynomial of degree $m+1$.