Diophantine geometry over groups. I: Makanin-Razborov diagrams
type: Item
instance of: scholarly article
title: Diophantine geometry over groups. I: Makanin-Razborov diagrams
zbMATH Open document ID: 1018.20034
published in: Publications Mathématiques
publication date: 2003-03-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0609096
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104176
review text: In this paper the author starts a sequence that borrows concepts and techniques from geometric group theory, low dimensional topology, and Diophantine geometry to study the structure of varieties defined over a free group. The canonical Makanin-Razborov diagram that encodes the set of solutions of a system of equations over a free group is presented. Parametric families of sets of solutions, and associated with such families canonical graded Makanin-Razborov diagrams are discussed. This first paper starts by studying limit groups that are obtained from Gromov limits of sequences of homomorphisms. Then the canonical Abelian JSJ-decompositions of these groups are studied. It is followed by consideration of the canonical cyclic JSJ-decomposition of a limit group. The canonical cyclic JSJ-decompositions are used to associate an analysis lattice with a limit group. It follows that limit groups are finitely presented, and a finitely generated group is a limit group if and only if it is \(\omega\)-residually free. A canonical Makanin-Razborov diagram is associated with a limit group. The class of finitely generated groups elementary equivalent to a nonabelian free group is described. The graded Makanin-Razborov diagram is introduced to study graded limit groups and systems of equations with parameters. Thus graded and multi-graded limit groups are the basic objects used by the author to study elementary sets defined over a free group.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 1874406
zbMATH Keywords: free groups
zbMATH Keywords: equations over groups
zbMATH Keywords: Diophantine geometry over groups
zbMATH Keywords: Makanin-Razborov diagrams
zbMATH Keywords: limit groups
zbMATH Keywords: JSJ-decompositions
zbMATH Keywords: finitely generated groups
zbMATH Keywords: residually free groups
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
OpenAlex ID: W1981155305
arXiv ID: math/0609096
DOI: 10.1007/S10240-001-8188-Y
author: Zlil Sela
description: scientific article