Diophantine geometry over groups. I: Makanin-Razborov diagrams
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.
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