[Paper Review] Regular completions of Z^n-free groups

Olga Kharlampovich, Alexei Miasnikov+1 more|arXiv (Cornell University)|Aug 22, 2012

Geometric and Algebraic Topology17 references3 citations

TL;DR

This paper introduces a method to embed any finitely generated Z^n-free group G into a larger finitely generated Z^n-free group H that acts regularly on a Z^n-tree, preserving G's action. The construction is effective when G is represented as Z^n-words, yielding an effective word representation for H as well.

ABSTRACT

In the present paper we continue studying regular free group actions on Z^n-trees. We show that every finitely generated Z^n-free group G can be embedded into a finitely generated Z^n-free group H acting regularly on the underlying Z^n-tree (we call H a regular Z^n-completion of G) so that the action of G is preserved. Moreover, if G is effectively represented as a group of Z^n-words then the construction of H is effective and H is effectively represented as a group of Z^n-words.

Motivation & Objective

• To develop a method for embedding finitely generated Z^n-free groups into larger Z^n-free groups with regular actions on Z^n-trees.
• To ensure the embedding preserves the original group action on the underlying Z^n-tree.
• To provide an effective construction when the original group is represented as Z^n-words.
• To demonstrate that the resulting completion group H is effectively representable as a group of Z^n-words.

Proposed method

• The construction uses a systematic extension of the group G to a larger group H by adding generators and relations to ensure regularity of the action on a Z^n-tree.
• The method relies on the structure of Z^n-trees and the properties of Z^n-free groups to maintain the group's free action with trivial arc stabilizers.
• The embedding is designed so that the action of G on the Z^n-tree is isomorphic to its image in H, preserving the original dynamics.
• When G is given as a group of Z^n-words, the construction is effective, meaning the word problem for H is solvable and H admits a computable presentation.
• The method ensures H is finitely generated and remains Z^n-free, satisfying the required algebraic and geometric constraints.

Experimental results

Research questions

• RQ1 Can every finitely generated Z^n-free group be embedded into a Z^n-free group acting regularly on a Z^n-tree?
• RQ2 Is such an embedding effective when the original group is given as a group of Z^n-words?
• RQ3 Can the resulting completion group H be effectively represented as a group of Z^n-words?
• RQ4 Does the construction preserve the original group action on the Z^n-tree?

Key findings

• Every finitely generated Z^n-free group G admits a regular Z^n-completion H that acts regularly on a Z^n-tree.
• The action of G on the Z^n-tree is preserved under the embedding into H.
• When G is effectively represented as a group of Z^n-words, the completion H is also effectively representable as a group of Z^n-words.
• The construction of H is algorithmic and maintains the Z^n-freeness and finite generation of the original group.