Hierarchy for groups acting on hyperbolic $\mathbf{Z}^n$-spaces

TL;DR

This paper explores the structure of finitely generated groups acting on hyperbolic spaces with a $ extbf{Z}^n$-metric, establishing a hierarchical framework akin to that for $ extbf{Z}^n$-free groups, advancing understanding of such group actions.

Contribution

It introduces a hierarchical structure for groups acting on hyperbolic $ extbf{Z}^n$-spaces, extending previous work on $ extbf{Z}^n$-free groups to a broader setting.

Findings

Describes the structure of groups acting on hyperbolic $ extbf{Z}^n$-spaces.

Establishes a hierarchy similar to that for $ extbf{Z}^n$-free groups.

Provides conditions under which the hierarchy applies.

Abstract

In their first article, the authors initiated a systematic study of hyperbolic $\Lambda$-metric spaces, where $\Lambda$ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case $\Lambda = \mathbf{Z}^n$ taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic $\mathbf{Z}^n$-metric spaces. Under certain constraints, the structure of such groups is described in terms of a {\em hierarchy} similar to the one established for $\mathbf{Z}^n$-free groups by Kharlampovich, Myasnikov, Remeslennikov and Serbin.