Diophantine geometry over groups I : Makanin-Razborov diagrams

2001/09/07 by Zlil Sela · 7 citations
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.1007/s10240-001-8188-y

crossref issued 2001/09/07 · crossref published 2001/09/07 · crossref published-online 2001/09/07 · openalex publication_date 2001/09/07 · crossref created 2003/03/24 · openalex created_date 2025/10/10 · crossref deposited 2026/02/17 · openalex updated_date 2026/07/23 · crossref indexed 2026/08/02

Abstract

This paper is the first in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the first paper we present the (canonical) Makanin-Razborov diagram that encodes the set of solutions of a system of equations. We continue by studying parametric families of sets of solutions, and associate with such a family a canonical graded Makanin-Razborov diagram, that encodes the collection of Makanin-Razborov diagrams associated with the individual members in the parametric family.

Citations

• JSJ-splittings for finitely presented groups over slender groups
• Acylindrical accessibility for groups

Cited by

• Diophanting geometry over groups II: Completions, closures and formal solutions
• The isomorphism problem for toral relatively hyperbolic groups
• Word Measures on Unitary Groups
• Finitely presented residually free groups
• Elementary theory of free non-abelian groups
• DIOPHANTINE GEOMETRY OVER GROUPS VI: THE ELEMENTARY THEORY OF A FREE GROUP
• Zlil Sela [wikipedia]