Metric Behaviour of the Magnus Embedding

TL;DR

This paper demonstrates that the Magnus embedding is 2-bi-Lipschitz and uses this property to establish lower bounds on $L_p$ compression exponents in free solvable groups, advancing understanding of their metric properties.

Contribution

It provides a geometric, 2-bi-Lipschitz characterization of the Magnus embedding and applies this to analyze metric embeddings of free solvable groups.

Findings

Magnus embedding is 2-bi-Lipschitz with respect to natural generating sets.

Established a non-zero lower bound on $L_p$ compression exponents in free solvable groups.

Enhanced understanding of the metric behavior of the Magnus embedding in group theory.

Abstract

The classic Magnus embedding is a very effective tool in the study of abelian extensions of a finitely generated group $G$, allowing us to see the extension as a subgroup of a wreath product of a free abelian group with $G$. In particular, the embedding has proved to be useful when studying free solvable groups. An equivalent geometric definition of the Magnus embedding is constructed and it is used to show that it is 2-bi-Lipschitz, with respect to an obvious choice of generating sets. This is then applied to obtain a non-zero lower bound on $L_p$ compression exponents in free solvable groups.