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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 = ℤ n taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ℤ n -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- 2 0 9) , Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ℤ n -free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ℤ n , Trans. Amer. Math. Soc. 364 (2012) 2847–2882]. [ABSTRACT FROM AUTHOR]

In this paper, we consider the conjugacy stability property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free hyperbolic groups and finitely generated subgroups of limit groups. [ABSTRACT FROM AUTHOR]

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]

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]

Using graph-theoretic techniques for f.g. subgroups of Fℤ[t] we provide a criterion for a f.g. subgroup of a f.g. fully residually free group to be of finite index. Moreover, we show that this criterion can be checked effectively. As an application we obtain an analogue of Greenberg-Stallings Theorem for f.g. fully residually free groups, and prove that a f.g. nonabelian subgroup of a f.g. fully residually free group is of finite index in its normalizer and commensurator. [ABSTRACT FROM AUTHOR]

In this paper we study exponential extensions of G which are finitely generated G-subgroups of the [ t]-completion G[ t] of a given CSA-group G. Using BassSerre theory we prove that exponential extensions of G can be obtained from subgroups of G by free constructions of a special type. As an application of this technique we describe the cohomological and homological dimensions of finitely generated fully residually free groups and give an algorithm to compute them. [ABSTRACT FROM AUTHOR]

In this paper we discuss a general approach to positive theories of groups. As an application we get a robust description of positive theories of groups with regular free Lyndon length function. Our approach combines techniques of infinite words (see [17, 3]), cancellation diagrams introduced in [14], and Merzlyakov's method [15]. [ABSTRACT FROM AUTHOR]

Let F = F(X) be a free group with basis X and ℤ[t] be a ring of polynomials with integer coefficients in t. In this paper we develop a theory of (ℤ[t],X)-graphs — a powerful tool in studying finitely generated fully residually free (limit) groups. This theory is based on the Kharlampovich–Myasnikov characterization of finitely generated fully residually free groups as subgroups of the Lyndon's group Fℤ[t], the author's representation of elements of Fℤ[t] by infinite (ℤ[t],X)-words, and Stallings folding method for subgroups of free groups. As an application, we solve the membership problem for finitely generated subgroups of Fℤ[t], as well as for finitely generated fully residually free groups. [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]