Regular completions of $\mathbb{Z}^n$-free groups

TL;DR

This paper proves that any finitely generated $Z^n$-free group can be embedded into a finitely generated regular $Z^n$-free group, preserving the action and allowing effective construction via $Z^n$-words.

Contribution

It introduces the concept of regular $Z^n$-completion, showing how to embed groups into regular actions while maintaining effectiveness.

Findings

Every finitely generated $Z^n$-free group can be embedded into a regular $Z^n$-free group.

The construction of the regular completion is effective and preserves the group's $Z^n$-word representation.

The embedding preserves the original group's action on the $Z^n$-tree.

Abstract

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