[Paper Review] On the Magnus Embedding and the Conjugacy Length Function of Wreath Products and Free Solvable Groups

Andrew W. Sale|arXiv (Cornell University)|Feb 23, 2012

Geometric and Algebraic Topology4 citations

TL;DR

This paper redefines the Magnus embedding geometrically and proves it is 2-bi-Lipschitz with respect to natural generating sets, enabling a non-zero lower bound on the $L_p$ compression exponent in free solvable groups. The result establishes a quantitative geometric property of these groups via their embedding into wreath products.

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.

Motivation & Objective

• To provide a geometric characterization of the classic Magnus embedding for abelian extensions of finitely generated groups.
• To establish the bi-Lipschitz property of the Magnus embedding with respect to natural generating sets.
• To apply the bi-Lipschitz property to derive quantitative geometric invariants in free solvable groups.
• To determine a non-zero lower bound on the $L_p$ compression exponent for free solvable groups.

Proposed method

• Construct an equivalent geometric definition of the Magnus embedding using group actions and generating sets.
• Define a metric structure on the wreath product to analyze distortion under the embedding.
• Prove that the Magnus embedding is 2-bi-Lipschitz by comparing word lengths in the source and target groups.
• Utilize the bi-Lipschitz property to bound the growth of the $L_p$ compression function in free solvable groups.
• Apply known results on compression exponents to deduce a positive lower bound in the context of wreath product embeddings.

Experimental results

Research questions

• RQ1 Is the Magnus embedding 2-bi-Lipschitz with respect to standard generating sets of the wreath product?
• RQ2 Can the geometric formulation of the Magnus embedding yield quantitative geometric invariants in free solvable groups?
• RQ3 What is the $L_p$ compression exponent of free solvable groups, and can it be bounded below by a positive constant?
• RQ4 How does the bi-Lipschitz property of the Magnus embedding relate to the geometry of wreath products?

Key findings

• The Magnus embedding is proven to be 2-bi-Lipschitz with respect to an obvious choice of generating sets in the wreath product.
• A geometric reformulation of the Magnus embedding is constructed, providing a new perspective on its structure and distortion.
• The bi-Lipschitz property enables the derivation of a non-zero lower bound on the $L_p$ compression exponent for free solvable groups.
• The result confirms that free solvable groups exhibit non-trivial $L_p$ compression, indicating non-amenable-like geometric behavior.