Endomorphisms of nilpotent groups of finite rank

TL;DR

This paper establishes criteria for when endomorphisms of torsion-free nilpotent groups of finite rank are automorphisms, extending known results beyond finitely generated groups by analyzing induced maps on abelianisation and the centre.

Contribution

It generalizes existing automorphism criteria for finitely generated nilpotent groups to the broader class of finite rank groups, addressing new challenges.

Findings

Criteria for automorphisms based on induced maps

Extension of known results to non-finitely generated groups

Addresses difficulties in the finite rank case

Abstract

We obtain sufficient criteria for endomorphisms of torsion-free nilpotent groups of finite rank to be automorphisms, by considering the induced maps on the torsion-free abelianisation and the centre. Whilst these results are known in the finitely generated case removing this assumption introduces several difficulties.