arXiv subjects

Zlil Sela

Publications and source records attributed to Zlil Sela.

7 recordsLinked to original sources

arXiv2026

Diophantine Geometry over Groups IX: Envelopes and Imaginaries

This paper is the ninth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets in free and hyperbolic groups. In the ninth paper we associate a Diophantine set with a definable set, and view it as the Diophantine envelope of the definable set. We use the envelope and duo limit groups that were used in proving stability of the theory of free and torsion-free hyperbolic groups [Se9], to study definable equivalence relations, and in particular, to classify imaginaries in these groups.

math.GR↗

arXiv2024

Equations in Burnside groups

This article is a first step in the study of equations in periodic groups. As an application, we study the structure of periodic quotients of hyperbolic groups. We investigate for instance the Hopf and co-Hopf properties, the isomorphism problem, the existence of free splittings, etc. We also consider the automorphism group of such periodic groups.

math.GR↗

arXiv2023

The rates of growth in a hyperbolic group

We study the countable set of rates of growth of a hyperbolic group with respect to all its finite generating sets. We prove that the set is well-ordered, and that every real number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. We prove that the ordinal of the set of rates of growth is at least $ω^ω$, and in case the group is a limit group (e.g., free and surface groups), it is $ω^ω$. We further study the rates of growth of all the finitely generated subgroups of a hyperbolic group with respect to all their finite generating sets. This set is proved to be well-ordered as well, and every real number can be the rate of growth of at most finitely many isomorphism classes of finite generating sets of subgroups of a given hyperbolic group. Finally, we strengthen our results to include rates of growth of all the finite generating sets of all the subsemigroups of a hyperbolic group.

math.GR↗

arXiv2019

Non Commutative Algebraic Geometry I: Monomial Equations with a Single Variable

This paper is the first in a sequence on the structure of sets of solutions to systems of equations over a free associative algebra. We start by constructing a Makanin-Razborov diagram that encodes all the homogeneous solutions to a homogeneous system of equations. Then we analyze the set of solutions to monomial systems of equations with a single variable.

math.RA↗

arXiv2012

Diophantine Geometry over Groups VIII: Stability

This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In the eighth paper we use a modification of the sieve procedure, that was presented as part of the quantifier elimination process, to prove that free and torsion-free (Gromov) hyperbolic groups are stable.

math.GR↗

arXiv2010

Diophantine Geometry over Groups X: The Elementary Theory of Free Products of Groups

This paper is the 10th in a sequence on the structure of sets of solutions to systems of equations over groups, projections of such sets (Diophantine sets), and the structure of definable sets over few classes of groups. In the 10th paper we answer affirmatively a question of R. L. Vaught on the elementary equivalence of free products of pairs of elementarily equivalent groups, and obtain a generalization of Tarski's problem on the elementary equivalence of non-abelian free groups. Finally, we prove that free products of stable groups is stable, generalizing a previous result on the stability of free groups.

math.GR↗

arXiv2003

Diophantine geometry over groups and the elementary theory of free and hyperbolic groups

We study sets of solutions to equations over a free group, projections of such sets, and the structure of elementary sets defined over a free group. The structre theory we obtain enable us to answer some questions of A. Tarski's, and classify those finitely generated groups that are elementary equivalent to a free group. Connections with low dimensional topology, and a generalization to (Gromov) hyperbolic groups will also be discussed.

math.GR↗