[Paper Review] Groups acting freely on $Λ$-trees

Olga Kharlampovich, Alexei Myasnikov+1 more|arXiv (Cornell University)|Nov 1, 2009

Geometric and Algebraic Topology23 references3 citations

TL;DR

This paper establishes that every finitely presented group with a free Lyndon length function over an ordered abelian group Λ can be embedded into a finitely presented group with a regular free length function in Λ, preserving the original length function. Furthermore, such groups with regular free length functions in Λ are shown to embed into ℝⁿ with lexicographic ordering and arise via a finite sequence of HNN-extensions with maximal abelian associated subgroups and length-isomorphic isomorphisms.

ABSTRACT

A group is called $Λ$-free if it has a free Lyndon length function in an ordered abelian group $Λ$, which is equivalent to having a free isometric action on a $Λ$-tree. A group has a regular free length function in $Λ$ if and only if it has a free isometric action on a $Λ$-tree so that all branch points belong to the orbit of the base point. In this paper we prove that every finitely presented $Λ$-free group $G$ can be embedded into a finitely presented group with a regular free length function in $Λ$ so that the length function on $G$ is preserved by the embedding. Next, we prove that every finitely presented group $\widetilde G$ with a regular free Lyndon length function in $Λ$ has a regular free Lyndon length function in ${\mathbb R}^n$ ordered lexicographically for an appropriate $n$ and can be obtained from a free group by a series of finitely many HNN-extensions in which associated subgroups are maximal abelian and length isomorphic.

Motivation & Objective

• To solve a key part of the Alperin-Bass program by characterizing finitely presented Λ-free groups for arbitrary ordered abelian groups Λ.
• To show that every finitely presented Λ-free group admits a length-preserving embedding into a group with a regular free length function in Λ.
• To prove that finitely presented groups with regular free length functions in Λ are isomorphic to groups obtained by a finite sequence of HNN-extensions with maximal abelian associated subgroups and length-isomorphic isomorphisms.
• To establish that such groups can be realized as acting freely and regularly on ℝⁿ-trees with lexicographic ordering.
• To bridge the gap between abstract length functions and geometric actions on Λ-trees by introducing regularity and completeness in the Lyndon length function framework.

Proposed method

• Use of generalized equations and the elimination process to analyze the structure of groups acting on Λ-trees.
• Application of elementary transformations in the context of linear elimination processes to control infinitesimal lengths and preserve the length function.
• Construction of a group embedding via the Rips machine and foliated band complexes with measured foliations in Λ.
• Reduction of the length function from Λ to ℝⁿ with lexicographic ordering using a decomposition of the ordered abelian group into a direct sum of ℝ-subgroups.
• Employment of HNN-extensions with maximal abelian associated subgroups and length-isomorphic isomorphisms to build the target group structure.
• Use of induction on the number of levels in the elimination process to prove regularity and invariance of the length function under redefinition in ℝⁿ.

Experimental results

Research questions

• RQ1 Can every finitely presented Λ-free group be embedded into a finitely presented group with a regular free length function in Λ while preserving the original length function?
• RQ2 What is the structural characterization of finitely presented groups with regular free Lyndon length functions in an arbitrary ordered abelian group Λ?
• RQ3 How can such groups be constructed from free groups using HNN-extensions with specific properties?
• RQ4 Is it possible to realize a Λ-free group action on a Λ-tree as a regular action on an ℝⁿ-tree with lexicographic ordering?
• RQ5 What role does the regularity condition play in ensuring that length functions in Λ can be realized via HNN-extensions with maximal abelian associated subgroups?

Key findings

• Every finitely presented Λ-free group can be embedded into a finitely presented group with a regular free length function in Λ, preserving the original length function.
• A finitely presented group with a regular free Lyndon length function in Λ admits a regular free length function in ℝⁿ with lexicographic ordering for some n.
• Such groups are constructed as a finite sequence of HNN-extensions where associated subgroups are maximal abelian and the isomorphisms are length-isomorphic.
• The length function in ℝⁿ coincides with the Lyndon length function after redefinition, and the action is free and regular on the corresponding ℝⁿ-tree.
• The structure of the group is fully determined by the elimination process and the resulting coordinate group of the generalized equation, with rank preserved under elementary transformations.
• The regularity of the length function ensures that elements of infinitesimal length in the original Λ must lie in a subgroup Λ′′, which is preserved under the embedding process.