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In this paper we study group actions on hyperbolic Λ-metric spaces, where Λ is an ordered abelian group. Λ-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 hyperbolicity for such spaces. Only the case of 0-hyperbolic Λ-metric spaces (that is, Λ-trees) was systematically studied, while the theory of general hyperbolic Λ-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 Λ-metric spaces for an arbitrary Λ. The other goal was to show some principal difficulties which arise in this generalization and the ways to deal with them. [ABSTRACT FROM AUTHOR]

In this paper we survey recent developments in the theory of groups acting on A-trees. We are trying to unify all significant methods and techniques, both classical and recently developed, in an attempt to present various faces of the theory and to show how these methods can be used to solve major problems about finitely presented A-free groups. Besides surveying results known up to date we draw many new corollaries concerning structural and algorithmic properties of such groups. [ABSTRACT FROM AUTHOR]

Abstract: Groups acting freely on -trees (-free groups) play a key role in the study of non-archimedean group actions. Following Stallingsʼ ideas, we develop graph-theoretic techniques to investigate subgroup structure of -free groups. As an immediate application of the presented method, we give an effective solution to the Uniform Membership Problem and the Power Problem in -free groups. [Copyright &y& Elsevier]

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Zn-trees give one a powerful tool to study groups. All finitely generated groups acting freely on R-trees also act freely on some Zn-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for Zn-actions (which is not the case for R-trees) comes along with a robust algorithmic theory. In this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on Zn-trees and give necessary and sufficient conditions for such actions. [ABSTRACT FROM AUTHOR]

This is the first paper in a series of three where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Our main goal is to show that group actions on Zn-trees give one a powerful tool to study groups. All finitely generated groups acting freely on R-trees also act freely on some Zn-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for Zn-actions (which is not the case for R-trees) comes along with a robust algorithmic theory. In this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on Zn-trees and give necessary and sufficient conditions for such actions. [ABSTRACT FROM AUTHOR]