  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.

Group Theory