Distortion in Free Nilpotent Groups

TL;DR

This paper characterizes undistorted subgroups in finitely generated free nilpotent groups, showing they are precisely the retracts of finite index subgroups, thus providing a clear criterion for subgroup distortion.

Contribution

It establishes a complete characterization of undistorted subgroups in free nilpotent groups as retracts of finite index subgroups, a novel result in geometric group theory.

Findings

Undistorted subgroups are exactly the retracts of finite index subgroups.

Provides a criterion for subgroup distortion in free nilpotent groups.

Advances understanding of subgroup geometry in nilpotent groups.

Abstract

We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.