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

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 $m+1$.

💡 Research Summary

The paper investigates the conjugator length function CL(n) in finitely generated 2‑step nilpotent groups. For a group G with a finite generating set, CL(n) is defined as the maximal length of a shortest conjugating word between any two conjugate elements whose total word length does not exceed n. The authors first establish a general polynomial upper bound for CL(n) that depends only on the rank m of the central subgroup Z in a central extension 1→Z→G→A→1, where A is abelian and Z≅ℤ^m×T (T finite).

Using a normal form that writes each element uniquely as a product of generators a_i (lifting a basis of A) followed by central generators c_j, they translate the conjugacy condition into a system of linear Diophantine equations M x = b. The matrix M encodes commutator coefficients γ_{ij} and the orders of the finite cyclic factors of Z. By carefully bounding the entries of M (|λ_{ij}|≤k L n, where L is a bound on the γ’s) and applying a lower‑triangular change of basis P∈SL_d(ℤ), they obtain a modified matrix M′ whose lower‑left block consists of residues modulo the finite orders.

The crucial step is the application of a theorem of Borosh, Flahive, Rubin, and Treybig (1989), which guarantees that any integer solution of a full‑rank linear system K x = b can be chosen with coordinates bounded by the maximal absolute value of the r×r minors of the augmented matrix