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

TL;DR

The paper studies the conjugacy problem in 2-step nilpotent groups by converting conjugacy to a system of linear Diophantine equations via normal forms and applying a BFRT-type bound to control integral solutions. It proves a general upper bound $CL(n) \preceq n^{m+1}$ for finitely generated class-2 nilpotent central extensions with center $Z \cong \mathbb{Z}^m \times$ (torsion), and constructs the family of groups $G_m$ to show matching lower bounds $CL(n) \succeq n^{m+1}$, establishing sharp polynomial growth of degree $m+1$ for the conjugator length. This reveals a divergence between the complexity of the word problem and the conjugacy problem in class-2 nilpotent groups and demonstrates how Diophantine techniques can tightly bound conjugator lengths. The results extend the understanding of how central extensions influence conjugacy and provide a framework for constructing groups with prescribed conjugator length behavior.

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 $m+1$.

Theorems & Definitions (5)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4: BFRT-Diophantine
• proof : Proof of Theorem \ref{['central extensions CL lower bound examples']}