# Diophantine geometry over groups I : Makanin-Razborov diagrams

Publications Mathématiques de l'IHÉS (2001)

- Volume: 93, page 31-105
- ISSN: 0073-8301

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topSela, Zlil. "Diophantine geometry over groups I : Makanin-Razborov diagrams." Publications Mathématiques de l'IHÉS 93 (2001): 31-105. <http://eudml.org/doc/104176>.

@article{Sela2001,

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.},

author = {Sela, Zlil},

journal = {Publications Mathématiques de l'IHÉS},

keywords = {free groups; equations over groups; Diophantine geometry over groups; Makanin-Razborov diagrams; limit groups; JSJ-decompositions; finitely generated groups; residually free groups},

language = {eng},

pages = {31-105},

publisher = {Institut des Hautes Etudes Scientifiques},

title = {Diophantine geometry over groups I : Makanin-Razborov diagrams},

url = {http://eudml.org/doc/104176},

volume = {93},

year = {2001},

}

TY - JOUR

AU - Sela, Zlil

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

JO - Publications Mathématiques de l'IHÉS

PY - 2001

PB - Institut des Hautes Etudes Scientifiques

VL - 93

SP - 31

EP - 105

AB - 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.

LA - eng

KW - free groups; equations over groups; Diophantine geometry over groups; Makanin-Razborov diagrams; limit groups; JSJ-decompositions; finitely generated groups; residually free groups

UR - http://eudml.org/doc/104176

ER -

## References

top- [Be] M. BESTVINA, R-trees in topology, geometry, and group theory, preprint. Zbl0998.57003MR1886668
- [Be-Fe1] M. BESTVINA and M. FEIGHN, Stable actions of groups on real trees, Inventiones Math. 121 (1995), 287-321. Zbl0837.20047MR1346208
- [Be-Fe2] M. BESTVINA and M. FEIGHN, Bounding the complexity of simplicial group actions, Inventiones Math. 103 (1991), 449-469. Zbl0724.20019MR1091614
- [De-Po] T. DELZANT and L. POTYAGAILO, Accessibilité hiérarchique, Topology, to appear. MR1838998
- [Du] M. DUNWOODY, Groups acting on protrees, J. London Math. Soc. 56 (1997), 125-136. Zbl0918.20011MR1462830
- [Du-Sa] M. DUNWOODY and M. SAGEEV, JSJ-splittings for finitely presented groups over slender groups, Inventiones Math. 135 (1999), 25-44. Zbl0939.20047MR1664694
- [Fa] B. FARB, Relatively hyperbolic groups, GAFA 8 (1998), 810-840. Zbl0985.20027MR1650094
- [Fu-Pa] K. FUJIWARA and P. PAPASOGLU, JSJ decompositions and complexes of groups, preprint. Zbl1097.20037
- [Gu] V. S. GUBA, Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Math. Zametki 40 (1986), 321-324. Zbl0611.20020MR869922
- [Kh-My] O. KHARLAMPOVICH and A. MYASNIKOV, Irreducible affine varieties over a free group II, J. of Algebra 200 (1998), 517-570. Zbl0904.20017MR1610664
- [Ma1] G. S. MAKANIN, Equations in a free group, Math. USSR Izvestiya 21 (1983), 449-469. Zbl0527.20018MR682490
- [Ma2] G. S. MAKANIN, Decidability of the universal and positive theories of a free group, Math. USSR Izvestiya 25 (1985), 75-88. Zbl0578.20001MR755956
- [Me] Yu. I. MERZLYAKOV, Positive formulae on free groups, Algebra i Logika 5 (1966), 257-266. MR222149
- [Pa] F. PAULIN, Outer automorphisms of hyperbolic groups and small actions on R-trees, Arboreal Group Theory (ed. R. C. Alperin), 331-343. Zbl0804.57002MR1105339
- [Ra1] A. A. RAZBOROV, On systems of equations in a free group, Math. USSR Izvestiya 25 (1985), 115-162. Zbl0579.20019MR755958
- [Ra2] A. A. RAZBOROV, On systems of equations in a free group, Ph.D. thesis, Steklov Math. institute (1987).
- [Ri-Se1] E. RIPS and Z. SELA, Structure and rigidity in hyperbolic groups I, GAFA 4 (1994), 337-371. Zbl0818.20042MR1274119
- [Ri-Se2] E. RIPS and Z. SELA, Cyclic splittings of finitely presented groups and the canonical JSJ decomposition, Annals of Mathematics 146 (1997), 53-104. Zbl0910.57002MR1469317
- [Se1] Z. SELA, The Nielsen-Thurston classification and automorphisms of a free group I, Duke Math. J. 84 (1996), 379-397. Zbl0858.20019MR1404334
- [Se2] Z. SELA, Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups II, GAFA 7 (1997), 561-593. Zbl0884.20025MR1466338
- [Se3] Z. SELA, Acylindrical accessibility for groups, Inventiones Mathematicae 129 (1997), 527-565. Zbl0887.20017MR1465334
- [Se4] Z. SELA, Endomorphisms of hyperbolic groups I: The Hopf property, Topology 38 (1999), 301-321. Zbl0929.20033MR1660337
- [We] R. WEIDMANN, The Nielsen method for groups acting on trees, preprint. Zbl1018.20020MR1901370

## Citations in EuDML Documents

top- Chloé Perin, Erratum to: “Elementary embeddings in torsion-free hyperbolic groups”
- Chloé Perin, Elementary embeddings in torsion-free hyperbolic groups
- Frédéric Paulin, Sur la théorie élémentaire des groupes libres
- François Dahmani, Daniel Groves, The isomorphism problem for toral relatively hyperbolic groups
- Vincent Guirardel, Actions of finitely generated groups on $\mathbb{R}$-trees

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