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Olga Kharlampovich

Publications and source records attributed to Olga Kharlampovich.

51 records · Page 3Linked to original sources
arXiv2013

Effective embedding of residually hyperbolic groups into direct products of extensions of centralizers

For any torsion-free hyperbolic group $Γ$ and any group $G$ that is fully residually $Γ$, we construct algorithmically a finite collection of homomorphisms from $G$ to groups obtained from $Γ$ by extensions of centralizers, at least one of which is injective. When $G$ is residually $Γ$, this gives a effective embedding of $G$ into a direct product of such groups. We also give an algorithmic construction of a diagram encoding the set of homomorphisms from a given finitely presented group to $Γ$.

math.GR↗
arXiv2013

Actions, length functions, and non-archemedian 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.

math.GR↗
arXiv2012

Definable sets in a hyperbolic group

We give a description of definable sets $P=(p_1,..., p_m)$ in a free non-abelian group $F$ and in a torsion-free non-elementary hyperbolic group $G$ that follows from our work on the Tarski problems. This answers Malcev's question for $F$. As a corollary we show that proper non-cyclic subgroups of $F$ and $G$ are not definable and prove Bestvina and Feighn's result that definable subsets $P=(p)$ in a free group are either negligible or co-negligible in their terminology.

math.GR↗
arXiv2011

From automatic structures to automatic groups

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy many nice properties: these group are invariant under the change of generators, they are closed under direct and free products, certain types of amalgamated products, and finite extensions. Furthermore, the Word Problem in graph automatic groups is decidable in quadratic time. However, the class of graph automatic groups is much wider then the class of automatic groups. For example, we prove that all finitely generated 2-nilpotent groups and Baumslag-Solitar groups B(1,n) are graph automatic, as well as many other metabelian groups.

math.GR↗
arXiv2011

Linear estimates for solutions of quadratic equations in free groups

We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by $2s$ for orientable equation and by $12s^4$ for non-orientable equation, where $s$ is the sum of the lengths of the coefficients

math.GR↗
arXiv2011

Infinite words and universal free actions

This is the second paper in a series of three, 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 $Γ_G$ equipped with a free action of $G$. Moreover, we show that $Γ_G$ is a universal tree for $G$ in the sense that it isometrically embeds in every $Λ$-tree equipped with a free $G$-action compatible with the original length function on $G$.

math.GR↗
arXiv2009

Groups with free regular length functions in Z^n

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 Z^n-trees give one a powerful tool to study groups. All finitely generated groups acting freely on -trees also act freely on some Z^n-trees, but the latter ones form a much larger class. The natural effectiveness of all constructions for $Z^n-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 Z^n-trees and give necessary and sufficient conditions for such actions.

math.GR↗
arXiv2009

Equations and fully residually free groups

This paper represents notes of the mini-courses given by the authors at the GCGTA conference in Dortmund (2007), Ottawa-Saint Sauveur conference (2007), Escola d'Algebra in Rio de Janeiro (2008) and Alagna (Italy, 2008) conference on equations in groups. We explain here the Elimination process for solving equations in a free group which has Makanin-Razborov process as a prototype. We also explain how we use this process to obtain the structure theorem for finitely generated fully residually free groups and many other results.

math.GR↗
arXiv2005

Effective JSJ Decompositions

In this paper we describe an elimination process which is a deterministic rewriting procedure that on each elementary step transforms one system of equations over free groups into a finitely many new ones. Infinite branches of this process correspond to cyclic splittings of the coordinate group of the initial system of equations. This allows us to construct algorithmically Grushko's decompositions of finitely generated fully residually free groups and cyclic [abelian] JSJ decompositions of freely indecomposable finitely generated fully residually free groups. We apply these results to obtain an effective description of the set of homomorphisms from a given finitely presented group into a free group, or, more generally, into an NTQ group.

math.GR↗
arXiv2005

Algebraic Geometry over Free Groups: Lifting Solutions into Generic Points

In this paper we prove Implicit Function Theorems (IFT) for algebraic varieties defined by regular quadratic equations and, more generally, regular NTQ systems over free groups. In the model theoretic language these results state the existence of very simple Skolem functions for particular $\forall\exists$-formulas over free groups. We construct these functions effectively. In non-effective form IFT first appeared in \cite{Imp}. From algebraic geometry view-point IFT can be described as lifting solutions of equations into generic points of algebraic varieties. Moreover, we show that the converse is also true, i.e., IFT holds only for algebraic varieties defined by regular NTQ systems. This implies that if a finitely generated group $H$ is $\forall\exists$-equivalent to a free non-abelian group then $H$ is isomorphic to the coordinate group of a regular NTQ system.

math.GR↗
arXiv2005

Isomorphism problem for finitely generated fully residually free groups

We prove that the isomorphism problem for finitely generated fully residually free groups is decidable. We also show that each finitely generated fully residually free group G has a decomposition that is invariant under automorphisms of G, and obtain a structure theorem for the group of outer automorphisms Out(G).

math.GR↗
arXiv1996

CSA groups and separated free constructions

A group $G$ is said to be a {\it CSA}-group if all maximal abelian subgroups of $G$ are malnormal. The class of CSA groups is of interest because it contains torsion-free hyperbolic groups, groups acting freely on $Λ$-trees and groups with the same existential theory as free groups. CSA groups are also very closely related to the study of residually free groups and tensor completions. In this paper we investigate which free constructions (amalgamated products and HNN extensions) over CSA groups are again CSA. The results are applied, in particular, to show that a torsion-free one-relator group is CSA if and only if it does not contain nonabelian metabelin Baumslag-Solitar groups and the direct product of the free group of rank 2 and the infinite cyclic group.

math.GR↗
arXiv1996

Equations in a free Q-group

In this work we investigate tensor completions of groups by associative rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main result states that there exists an algorithm that decides if a given finite system of equations over a free ${\bf Q}$-group has a solution, and if it does, finds a solution. This statement can be generalized for ${\bf Q}$-completions of torsion-free hyperbolic groups. Our proof significantly uses the techniques of word hyperbolic groups and the results of E.Rips and Z.Sela on the solvability of systems of equations in hyperbolic groups.

math.GR↗
arXiv1996

Hyperbolic groups and free constructions

We investigate which free constructions (amalgamated products and HNN-extensions) over word hyperbolic groups produce groups that are again word hyperbolic. A complete answer is obtained for the case when the amalgamated subgroups are virtually cyclic. The results are applied, in particular, to show that a ${\bf Q}$-completion of a torsion-free hyperbolic group has solvable word problem and conjugacy problem.

math.GR↗