arXiv subjects

Z. Sela

Publications and source records attributed to Z. Sela.

7 recordsLinked to original sources

arXiv2025

A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I

The JSJ decomposition and the Makanin-Razborov diagram were proved to be essential in studying varieties over free groups, semigroups and associative algebras. In this paper we suggest a unified conceptual approach to the applicability of these structures over all these algebraic categories. With a variety over each of these algebraic categories we naturally associate a set of tuples in a free group. Then we show how to associate a Makanin-Razborov diagram with any set of tuples over a free group. Furthermore, in case the MR diagram that is associated with a set of tuples is single ended, we prove that there is a canonical Makanin-Razborov diagram that can be associated with such a set. This canonical diagram is a main key in studying varieties over free semigroups, associative algebras and Lie algebras, and encodes the global structure of these varieties. It enables us to define a (pseudo) closure of a set of tuples over each of the algebraic objects, associate a rank with it (analogous to Shelah and Lascar ranks), and over free groups the closure provides a canonical envelope that is essential in studying the structure and the properties of definable sets.

math.GR↗

arXiv2023

Automorphisms of Groups and a Higher Rank JSJ Decomposition II: The single ended case

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group (HHG) that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. This higher rank decomposition encodes the dynamics of individual automorphisms and the structure of the outer automorphism group of an HHG.

math.GR↗

arXiv2023

Automorphisms of Groups and a Higher Rank JSJ Decomposition I: RAAGs and a Higher Rank Makanin-Razborov Diagram

The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct what we view as a higher rank Makanin-Razborov diagram, which is the first step in the construction of the higher rank JSJ.

math.GR↗

arXiv2016

Word Equations I: Pairs and their Makanin-Razborov Diagrams

This paper is the first in a sequence on the structure of sets of solutions to systems of equations over a free semigroup. To describe the structure, we present a Makanin-Razborov diagram that encodes the set of solutions to such system of equations. In the sequel we show how this diagram, and the tools that are used in constructing it, can be applied to analyze fragments of the first order theory of free semigroups.

math.GR↗

arXiv2014

A report on Tarski's decidability problem

This paper contains a list of crucial mistakes and counterexamples to some of the main statements in the paper "Elementary theory of free nonabelian groups" by O. Kharlampovich and A. Myasnikov, which was published in the Journal of Algebra in June 2006.

math.GR↗

arXiv2012

Free and Hyperbolic Groups are not Equational

We give an example of a definable set in every free or torsion-free (non-elementary) hyperbolic group that is not in the Boolean algebra of equational sets. Hence, the theories of free and torsion-free (non-elementary) hyperbolic groups are not equational in the sense of G. Srour.

math.LO↗

arXiv2009

Makanin-Razborov Diagrams over Free Products

This paper is the first in a sequence on the first order theory of free products. In the first paper we generalize the analysis of systems of equations over free and (torsion-free) hyperbolic groups, and analyze system of equations over free products. To do that we introduce limit groups over the class of free products, and show that a finitely presented group has a canonical (finite) collection of maximal limit quotients. We further extend this finite collection and associate a Makanin-Razborov diagram over free products with a finitely presented group. This MR diagram encodes all the quotients of a given finitely presented group that are free products, all its homomorphisms into free products, and equivalently all the solutions to a given system of equations over a free product.

math.GR↗