Distortion of embeddings of a torsion-free finitely generated nilpotent group into a unitriangular group

TL;DR

This paper investigates how certain embeddings of torsion-free finitely generated nilpotent groups into unitriangular groups are distorted and introduces an efficient algorithm to compute this distortion.

Contribution

It provides a detailed analysis of embedding distortions and presents a polynomial time algorithm for calculating subgroup distortion within these groups.

Findings

Identifies specific distortion behaviors of embeddings into unitriangular groups

Develops a polynomial time algorithm for distortion computation

Enhances understanding of subgroup geometry in nilpotent groups

Abstract

In this paper we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $UT_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a given subgroup of $G$