The Magnus embedding is a quasi-isometry

TL;DR

This paper proves that the Magnus embedding, which maps free solvable groups into wreath products, preserves large-scale geometric structure by being a quasi-isometry, enhancing understanding of their geometric properties.

Contribution

The paper establishes that the Magnus embedding is a quasi-isometry, providing new insights into the geometric structure of free solvable groups.

Findings

Magnus embedding is a quasi-isometry

Embedding preserves large-scale geometry

Implications for geometric group theory

Abstract

We show that the Magnus embedding, which embeds the free solvable group of rank r and degree d into the wreath product of the free abelian group of rank r with the free solvable group of rank r and degree d-1, is a quasi-isometry.