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

*(English)*Zbl 1018.20034In 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.

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.

Reviewer: V.A.Roman’kov (Omsk)

##### MSC:

20F65 | Geometric group theory |

20E05 | Free nonabelian groups |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

03B25 | Decidability of theories and sets of sentences |

20E26 | Residual properties and generalizations; residually finite groups |

20E36 | Automorphisms of infinite groups |