Denis Serbin

ON POSITIVE THEORIES OF GROUPS WITH REGULAR FREE LENGTH FUNCTIONS

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 Merzlyakovs method [15].

FULLY RESIDUALLY FREE GROUPS AND GRAPHS LABELED BY INFINITE WORDS

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 F Z[t] , the author’s representation of elements of F Z[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 F Z[t], as well as for finitely generated fully residually free groups.

ACTIONS, LENGTH FUNCTIONS, AND NON-ARCHIMEDEAN WORDS

In this paper we survey recent developments in the theory of groups acting on Λ-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 Λ-free groups. Besides surveying results known up to date we draw many new corollaries concerning structural and algorithmic properties of such groups.

FINITE INDEX SUBGROUPS OF FULLY RESIDUALLY FREE GROUPS

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.

Infinite words and universal free actions

Abstract. This is the second paper in a series of four, where we take on the unified theory of non-Archimedean group actions, length functions and infinite words. Here, for an arbitrary group G of infinite words over an ordered abelian group Λ we construct a Λ-tree Γ G

Exponential extensions of groups

\Gamma _G

Groups acting on hyperbolic Λ-metric spaces

equipped with a free action of G. Moreover, we show that Γ G

Groups acting freely on

\Gamma _G

\Lambda

is a universal tree for G in the sense that it isometrically and equivariantly embeds into every Λ-tree equipped with a free G-action compatible with the original length function on G. Also, for a group G acting freely on a Λ-tree Γ we show how one can easily obtain an embedding of G into the set of reduced infinite words R(Λ,X)

-trees

R(\Lambda , X)