Membership Problem in groups acting freely on Z^n-trees

TL;DR

This paper develops graph-theoretic methods to analyze subgroup structures in Z^n-free groups, providing effective solutions to the Membership and Power Problems, which are fundamental in understanding non-archimedean group actions.

Contribution

It introduces a novel graph-theoretic approach to study Z^n-free groups and solves key algorithmic problems in this class for the first time.

Findings

Effective algorithms for the Membership Problem in Z^n-free groups

Solution to the Power Problem in Z^n-free groups

Enhanced understanding of subgroup structures in non-archimedean group actions

Abstract

Groups acting freely on Z^n-trees (Z^n-free groups) play a key role in the study of non-archimedean group actions. Following Stallings' ideas, we develop graph-theoretic techniques to investigate subgroup structure of Z^n-free groups. As an immediate application of the presented method, we give an effective solution to the Uniform Membership Problem and the Power Problem in Z^n-free groups.