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Olga Kharlampovich

Publications and source records attributed to Olga Kharlampovich.

At least 37 records · Page 2Linked to original sources
arXiv2019

Quantifier elimination algorithm to boolean combination of $\exists\forall$-formulas in the theory of a free group

It was proved by Sela and by the authors that every formula in the theory of a free group $F$ is equivalent to a boolean combination of $\exists\forall$-formulas. We also proved that the elementary theory of a free group is decidable (there is an algorithm given a sentence to decide whether this sentence belongs to $Th(F)$). In this paper we give an algorithm for reduction of a first order formula over a free group to an equivalent boolean combination of $\exists\forall$-formulas.

math.GR↗
arXiv2019

Fraïssé limits of limit groups

We modify the notion of a Fraïssé class and show that various interesting classes of groups, notably the class of nonabelian limit groups and the class of finitely generated elementary free groups, admit Fraïssé limits. Furthermore, we rediscover Lyndon's $\Z[t]$-exponential completions of countable torsion-free CSA groups, as Fraïssé limits with respect to extensions of centralizers. Dedicated to the memory of Charles Sims.

math.LO↗
arXiv2019

Bi-interpretability of Some Monoids with the Arithmetic and Applications

We will prove bi-interpretability of the arithmetic $\N = \langle N, +,\cdot, 0, 1\rangle$ and the weak second order theory of $\N$ with the free monoid $\mathbb{M}_X$ of finite rank greater than 1 and with a non-trivial partially commutative monoid with trivial center. This bi-interpretability implies that finitely generated submonoids of these monoids are definable. Moreover, any recursively enumerable language in the alphabet $X$ is definable in $\mathbb{M}_X$. Primitive elements, and, therefore, free bases are definable in the free monoid. It has the so-called QFA property, namely there is a sentence $ϕ$ such that every finitely generated monoid satisfying $ϕ$ is isomorphic to $\mathbb{M}_X$. The same is true for a partially commutative monoid without center. We also prove that there is no quantifier elimination in the theory of any structure that is bi-interpretable with $\mathbb N$ to any boolean combination of formulas from $Π_n$ or $Σ_n$.

math.LO↗
arXiv2018

Low complexity algorithms in knot theory

We show that the genus problem for alternating knots with $n$ crossings has linear time complexity and is in Logspace$(n)$. Almost all alternating knots of given genus possess additional combinatorial structure, we call them standard. We show that the genus problem for these knots belongs to $TC^0$ circuit complexity class. We also show, that the equivalence problem for such knots with $n$ crossings has time complexity $n\log (n)$ and is in Logspace$(n)$ and $TC^{0}$ complexity classes.

math.GT↗
arXiv2017

Beyond Serre's "Trees" in two directions: $Λ$--trees and products of trees

Serre in "Trees" laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat--Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from "Trees": the theory of isometric group actions on $Λ$-trees and the theory of lattices in the product of trees where we describe in more detail results on arithmetic groups acting on a product of trees.

math.GR↗
arXiv2017

Undecidability of Equations in Free Lie Algebras

In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero.

math.LO↗
arXiv2017

Undecidability of the first order theories of free non-commutative Lie algebras

Let $R$ be a commutative integral unital domain and $L$ a free non-commutative Lie algebra over $R$. In this paper we show that the ring $R$ and its action on $L$ are 0-interpretable in $L$, viewed as a ring with the standard ring language $+, \cdot,0$. Furthermore, if $R$ has characteristic zero then we prove that the elementary theory $Th(L)$ of $L$ in the standard ring language is undecidable. To do so we show that the arithmetic ${\bf N} = \langle{\bf N}, +,\cdot,0 \rangle$ is 0-interpretable in $L$. This implies that the theory of $Th(L)$ has the independence property. These results answer some old questions on model theory of free Lie algebras.

math.LO↗
arXiv2017

Stallings graphs for quasi-convex subgroups

We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically associated with the corresponding subgroups. We show that this notion of Stallings graphs allows a unified approach to many algorithmic problems: some which had already been solved like the generalized membership problem or the computation of a quasi-convexity constant (Kapovich, 1996); and others such as the computation of intersections, the conjugacy or the almost malnormality problems. Our results extend earlier algorithmic results for the more restricted class of virtually free groups. We also extend our construction to relatively quasi-convex subgroups of relatively hyperbolic groups, under certain additional conditions.

math.GR↗
arXiv2016

Equations in Algebras

We show that the Diophantine problem(decidability of equations) is undecidable in free associative algebras over any field and in the group algebras over any field of a wide variety of torsion free groups, including toral relatively hyperbolic groups, right angled Artin groups, commutative transitive groups, the fundamental groups of various graph groups, etc.

math.LO↗
arXiv2016

Tarski-type problems for free associative algebras

In this paper we study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free associative algebras of finite rank over fields are elementarily equivalent if and only if their ranks are the same and the fields are equivalent in the weak second order logic. In particular, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. We prove that if an arbitrary ring $B$ with at least one Noetherian proper centralizer is first-order equivalent to a free associative algebra of finite rank over an infinite field then $B$ is also a free associative algebra of finite rank over a field. This solves the elementary classification problem for free associative algebras in a wide class of rings. Finally, we present a formula of the ring language which defines the set of free bases in a free associative algebra of finite rank.

math.LO↗
arXiv2015

Quadratic Equations in Hyperbolic Groups are NP-complete

We prove that in a torsion-free hyperbolic group $Γ$, the length of the value of each variable in a minimal solution of a quadratic equation $Q=1$ is bounded by $N|Q|^3$ for an orientable equation, and by $N|Q|^{4}$ for a non-orientable equation, where $|Q|$ is the length of the equation, and the constant $N$ can be computed. We show that the problem, whether a quadratic equation in $Γ$ has a solution, is in NP, and that there is a PSpace algorithm for solving arbitrary equations in $Γ$. If additionally $Γ$ is non-cyclic, then this problem (of deciding existence of a solution) is NP-complete. We also give a slightly larger bound for minimal solutions of quadratic equations in a toral relatively hyperbolic group.

math.GR↗
arXiv2014

On Tarski's Decidability Problem

This note provides a brief guide to the current state of the literature on Tarski's problems with emphasis on features that distinguish the approach based on combinatorial and algorithmic group theory from the topological approach to Tarski's problem. We use this note to provide corrections to some typos and to address some misconceptions from the recent report by Z. Sela about the relations between the concepts and results in the approaches to the Tarski problems. We were forced to read Sela's papers to be able to address some of his comments, and found errors in his papers 6, 3 and 4 on Diophantine Geometry published in GAFA and Israel J. Math. which we mention in Section 4. His proceedings of the ICM 2002 paper also contains wrong Theorem 6 (to make it correct one has to change the definition of non-elementary hyperbolic $ω$-residually free towers to make them equivalent to our coordinate groups of regular NTQ systems.)

math.GR↗
arXiv2013

SLP compression for solutions of equations with constraints in free and hyperbolic groups

The paper is a part of an ongoing program which aims to show that the existential theory in free groups (hyperbolic groups or even toral relatively hyperbolic) is NP-complete. For that we study compression of solutions with straight-line programs (SLPs) as suggested originally by Plandowski and Rytter in the context of a single word equation. We review some basic results on SLPs and give full proofs in order to keep this fundamental part of the program self-contained. Next we study systems of equations with constraints in free groups and more generally in free products of abelian groups. We show how to compress minimal solutions with extended Parikh-constraints. This type of constraints allows to express semi linear conditions as e.g. alphabetic information. The result relies on some combinatorial analysis and has not been shown elsewhere. We show similar compression results for Boolean formula of equations over a torsion-free $δ$-hyperbolic group. The situation is much more delicate than in free groups. As byproduct we improve the estimation of the "capacity" constant used by Rips and Sela in their paper "Canonical representatives and equations in hyperbolic groups" from a double-exponential bound in $δ$ to some single-exponential bound. The final section shows compression results for toral relatively hyperbolic group using the work of Dahmani: We show that given a system of equations over a fixed toral relatively hyperbolic group, for every solution of length $N$ there is an SLP for another solution such that the size of the SLP is bounded by some polynomial $p(s+ \log N)$ where $s$ is the size of the system.

math.GR↗
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↗