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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]

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]