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.

@article{PMIHES_2001__93__31_0, author = {Sela, Zlil}, title = {Diophantine geometry over groups I : Makanin-Razborov diagrams}, journal = {Publications Math\'ematiques de l'IH\'ES}, publisher = {Institut des Hautes \'Etudes Scientifiques}, volume = {93}, year = {2001}, pages = {31-105}, zbl = {1018.20034}, mrnumber = {1863735}, language = {en}, url = {http://www.numdam.org/item/PMIHES_2001__93__31_0} }

Sela, Zlil. Diophantine geometry over groups I : Makanin-Razborov diagrams. Publications Mathématiques de l'IHÉS, Volume 93 (2001) , pp. 31-105. http://www.numdam.org/item/PMIHES_2001__93__31_0/

[Be] R-trees in topology, geometry, and group theory, preprint. | MR 1886668 | Zbl 0998.57003

,[Be-Fe1] Stable actions of groups on real trees, Inventiones Math. 121 (1995), 287-321. | MR 1346208 | Zbl 0837.20047

and ,[Be-Fe2] Bounding the complexity of simplicial group actions, Inventiones Math. 103 (1991), 449-469. | MR 1091614 | Zbl 0724.20019

and ,[De-Po] T. DELZANT and L. POTYAGAILO, Accessibilité hiérarchique, Topology, to appear. | MR 1838998

[Du] Groups acting on protrees, J. London Math. Soc. 56 (1997), 125-136. | MR 1462830 | Zbl 0918.20011

,[Du-Sa] JSJ-splittings for finitely presented groups over slender groups, Inventiones Math. 135 (1999), 25-44. | MR 1664694 | Zbl 0939.20047

and ,[Fa] Relatively hyperbolic groups, GAFA 8 (1998), 810-840. | MR 1650094 | Zbl 0985.20027

,[Fu-Pa] JSJ decompositions and complexes of groups, preprint.

and ,[Gu] Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Math. Zametki 40 (1986), 321-324. | MR 869922 | Zbl 0611.20020

,[Kh-My] Irreducible affine varieties over a free group II, J. of Algebra 200 (1998), 517-570. | MR 1610664 | Zbl 0904.20017

and ,[Ma1] Equations in a free group, Math. USSR Izvestiya 21 (1983), 449-469. | MR 682490 | Zbl 0527.20018

,[Ma2] Decidability of the universal and positive theories of a free group, Math. USSR Izvestiya 25 (1985), 75-88. | MR 755956 | Zbl 0578.20001

,[Me] Yu. I. MERZLYAKOV, Positive formulae on free groups, Algebra i Logika 5 (1966), 257-266. | MR 222149

[Pa] Outer automorphisms of hyperbolic groups and small actions on R-trees, Arboreal Group Theory (ed. R. C. Alperin), 331-343. | MR 1105339 | Zbl 0804.57002

,[Ra1] On systems of equations in a free group, Math. USSR Izvestiya 25 (1985), 115-162. | MR 755958 | Zbl 0579.20019

,[Ra2] On systems of equations in a free group, Ph.D. thesis, Steklov Math. institute (1987).

,[Ri-Se1] Structure and rigidity in hyperbolic groups I, GAFA 4 (1994), 337-371. | MR 1274119 | Zbl 0818.20042

and ,[Ri-Se2] Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Annals of Mathematics 146 (1997), 53-104. | MR 1469317 | Zbl 0910.57002

and ,[Se1] The Nielsen-Thurston classification and automorphisms of a free group I, Duke Math. J. 84 (1996), 379-397. | MR 1404334 | Zbl 0858.20019

,[Se2] Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, GAFA 7 (1997), 561-593. | MR 1466338 | Zbl 0884.20025

,[Se3] Acylindrical accessibility for groups, Inventiones Mathematicae 129 (1997), 527-565. | MR 1465334 | Zbl 0887.20017

,[Se4] Endomorphisms of hyperbolic groups I: The Hopf property, Topology 38 (1999), 301-321. | MR 1660337 | Zbl 0929.20033

,[We] The Nielsen method for groups acting on trees, preprint. | MR 1901370 | Zbl 1018.20020

,