Diophantine geometry over groups I : Makanin-Razborov diagrams

Zlil Sela

Publications Mathématiques de l'IHÉS (2001)

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

Abstract

top
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.

How to cite

top

Sela, 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
  1. [Be] M. BESTVINA, R-trees in topology, geometry, and group theory, preprint. Zbl0998.57003MR1886668
  2. [Be-Fe1] M. BESTVINA and M. FEIGHN, Stable actions of groups on real trees, Inventiones Math. 121 (1995), 287-321. Zbl0837.20047MR1346208
  3. [Be-Fe2] M. BESTVINA and M. FEIGHN, Bounding the complexity of simplicial group actions, Inventiones Math. 103 (1991), 449-469. Zbl0724.20019MR1091614
  4. [De-Po] T. DELZANT and L. POTYAGAILO, Accessibilité hiérarchique, Topology, to appear. MR1838998
  5. [Du] M. DUNWOODY, Groups acting on protrees, J. London Math. Soc. 56 (1997), 125-136. Zbl0918.20011MR1462830
  6. [Du-Sa] M. DUNWOODY and M. SAGEEV, JSJ-splittings for finitely presented groups over slender groups, Inventiones Math. 135 (1999), 25-44. Zbl0939.20047MR1664694
  7. [Fa] B. FARB, Relatively hyperbolic groups, GAFA 8 (1998), 810-840. Zbl0985.20027MR1650094
  8. [Fu-Pa] K. FUJIWARA and P. PAPASOGLU, JSJ decompositions and complexes of groups, preprint. Zbl1097.20037
  9. [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
  10. [Kh-My] O. KHARLAMPOVICH and A. MYASNIKOV, Irreducible affine varieties over a free group II, J. of Algebra 200 (1998), 517-570. Zbl0904.20017MR1610664
  11. [Ma1] G. S. MAKANIN, Equations in a free group, Math. USSR Izvestiya 21 (1983), 449-469. Zbl0527.20018MR682490
  12. [Ma2] G. S. MAKANIN, Decidability of the universal and positive theories of a free group, Math. USSR Izvestiya 25 (1985), 75-88. Zbl0578.20001MR755956
  13. [Me] Yu. I. MERZLYAKOV, Positive formulae on free groups, Algebra i Logika 5 (1966), 257-266. MR222149
  14. [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
  15. [Ra1] A. A. RAZBOROV, On systems of equations in a free group, Math. USSR Izvestiya 25 (1985), 115-162. Zbl0579.20019MR755958
  16. [Ra2] A. A. RAZBOROV, On systems of equations in a free group, Ph.D. thesis, Steklov Math. institute (1987). 
  17. [Ri-Se1] E. RIPS and Z. SELA, Structure and rigidity in hyperbolic groups I, GAFA 4 (1994), 337-371. Zbl0818.20042MR1274119
  18. [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
  19. [Se1] Z. SELA, The Nielsen-Thurston classification and automorphisms of a free group I, Duke Math. J. 84 (1996), 379-397. Zbl0858.20019MR1404334
  20. [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
  21. [Se3] Z. SELA, Acylindrical accessibility for groups, Inventiones Mathematicae 129 (1997), 527-565. Zbl0887.20017MR1465334
  22. [Se4] Z. SELA, Endomorphisms of hyperbolic groups I: The Hopf property, Topology 38 (1999), 301-321. Zbl0929.20033MR1660337
  23. [We] R. WEIDMANN, The Nielsen method for groups acting on trees, preprint. Zbl1018.20020MR1901370

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.