Groups acting on hyperbolic $\Lambda$-metric spaces

In this paper we study group actions on hyperbolic $\Lambda$-metric spaces, where $\Lambda$ is an ordered abelian group. $\Lambda$-metric spaces were first introduced by Morgan and Shalen in their study of hyperbolic structures and then Chiswell, following Gromov's ideas, introduced the notion of hyperbolicty for such spaces. Only the case of 0-hyperbolic $\Lambda$-metric spaces (that is, $\Lambda$-trees) was systematically studied, while the theory of general hyperbolic $\Lambda$-metric spaces was not developed at all. Hence, one of the goals of the present paper was to fill this gap and translate basic notions and results from the theory of group actions on hyperbolic (in the usual sense) spaces to the case of $\Lambda$-metric spaces for an arbitrary $\Lambda$. The other goal was to show some principal difficulties which arise in this generalization and the ways to deal with them.