120 research outputs found

### On the Andrews-Curtis equivalence

The Andrews-Curtis conjecture claims that every balanced presentation of the
trivial group can be reduced to the standard one by a sequence of ``elementary
transformations" which are Nielsen transformations augmented by arbitrary
conjugations. It is a prevalent opinion that this conjecture is false; however,
not many potential counterexamples are known. In this paper, we show that some
of the previously proposed examples are actually not counterexamples. We hope
that the tricks we used in constructing relevant chains of elementary
transformations will be useful to those who attempt to establish the
Andrews-Curtis equivalence in other situations.
On the other hand, we give two rather general and simple methods for
constructing balanced presentations of the trivial group; some of these
presentations can be considered potential counterexamples to the Andrews-Curtis
conjecture. One of the methods is based on a simple combinatorial idea of
composition of group presentations, whereas the other one uses "exotic" knot
diagrams of the unknot.
We also consider the Andrews-Curtis equivalence in metabelian groups and
reveal some interesting connections of relevant problems to well-known problems
in K-theory.Comment: 16 pages, 1 figur

### Whitehead method and Genetic Algorithms

In this paper we discuss a genetic version (GWA) of the Whitehead's
algorithm, which is one of the basic algorithms in combinatorial group theory.
It turns out that GWA is surprisingly fast and outperforms the standard
Whitehead's algorithm in free groups of rank >= 5. Experimenting with GWA we
collected an interesting numerical data that clarifies the time-complexity of
the Whitehead's Problem in general. These experiments led us to several
mathematical conjectures. If confirmed they will shed light on hidden
mechanisms of Whitehead Method and geometry of automorphic orbits in free
groups.Comment: 29 pages, 7 figure

### Decidability of the Elementary Theory of a Torsion-Free Hyperbolic Group

Let G be a torsion free hyperbolic group. We prove that the elementary theory
of G is decidable and admits an effective quantifier elimination to boolean
combination of AE-formulas. The existence of such quantifier elimination was
previously proved by Sela.Comment: Misprints and errors corrected, referee suggestions addressed. arXiv
admin note: substantial text overlap with arXiv:1207.190

### A polynomial bound on solutions of quadratic equations in free groups

We provide polynomial upper bounds on the size of a shortest solution for
quadratic equations in a free group. A similar bound is given for parametric
solutions in the description of solutions sets of quadratic equations in a free
group.Comment: 41 pages, 10 figure

### 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.Comment: arXiv admin note: text overlap with arXiv:1606.0361

### 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
$\omega$-residually free towers to make them equivalent to our coordinate
groups of regular NTQ systems.)Comment: We address more comments in this versio

### 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

### 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

### 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.Comment: 126 pages, 15 figure

### 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.Comment: 33 pages, 7 figure

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