Hierarchy for groups acting on hyperbolic ℤn-spaces

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic Λ-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic Λ-metric spaces, where Λ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case Λ = ℤn taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ℤn-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy : (AMS-209), Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ℤn-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ℤn, Trans. Amer. Math. Soc. 364 (2012) 2847–2882].