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

In their first article, the authors initiated a systematic study of hyperbolic $\mathrm{\Lambda }$-metric spaces, where $\mathrm{\Lambda }$ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case $\mathrm{\Lambda }={\mathbf{Z}}^n$ 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.