Abstract
This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In this eighth paper we use a modification of the sieve procedure, which was used in proving quantifier elimination in the theory of a free group, to prove that free and torsion-free (Gromov) hyperbolic groups are stable.
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[Ly-Sch] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, New York: Springer-Verlag, 1977, vol. 89.
@book {Ly-Sch, MRKEY = {0577064},
AUTHOR = {Lyndon, Roger C. and Schupp, Paul E.},
TITLE = {Combinatorial Group Theory},
SERIES= {Ergeb. Math. Grenzgeb.},
VOLUME={89},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1977},
PAGES = {xiv+339},
ISBN = {3-540-07642-5},
MRCLASS = {20F05 (55A05)},
MRNUMBER = {0577064},
MRREVIEWER = {Ian M. Chiswell},
ZBLNUMBER = {0368.20023},
} -
[Pi] A. Pillay, An Introduction to Stability Theory, New York: The Clarendon Press Oxford University Press, 1983, vol. 8.
@book {Pi, MRKEY = {0719195},
AUTHOR = {Pillay, Anand},
TITLE = {An Introduction to Stability Theory},
SERIES = {Oxford Logic Guides},
VOLUME = {8},
PUBLISHER = {The Clarendon Press Oxford University Press},
ADDRESS = {New York},
YEAR = {1983},
PAGES = {xii+146},
ISBN = {0-19-853186-9},
MRCLASS = {03C45 (03-02)},
MRNUMBER = {0719195},
MRREVIEWER = {John T. Baldwin},
ZBLNUMBER = {0526.03014},
} -
[Pi-Sr]
A. Pillay and G. Srour, "Closed sets and chain conditions in stable theories," J. Symbolic Logic, vol. 49, iss. 4, pp. 1350-1362, 1984.@article {Pi-Sr, MRKEY = {0771800},
AUTHOR = {Pillay, Anand and Srour, Gabriel},
TITLE = {Closed sets and chain conditions in stable theories},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {49},
YEAR = {1984},
NUMBER = {4},
PAGES = {1350--1362},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03C45},
MRNUMBER = {0771800},
MRREVIEWER = {Daniel Lascar},
DOI = {10.2307/2274284},
ZBLNUMBER = {0597.03018},
} -
[Po1] B. Poizat, "Groupes stables, avec types génériques réguliers," J. Symbolic Logic, vol. 48, iss. 2, pp. 339-355, 1983.
@article {Po1, MRKEY = {0704088},
AUTHOR = {Poizat, Bruno},
TITLE = {Groupes stables, avec types génériques réguliers},
JOURNAL = {J. Symbolic Logic},
FJOURNAL = {The Journal of Symbolic Logic},
VOLUME = {48},
YEAR = {1983},
NUMBER = {2},
PAGES = {339--355},
ISSN = {0022-4812},
CODEN = {JSYLA6},
MRCLASS = {03C60 (03C45 20A15)},
MRNUMBER = {0704088},
MRREVIEWER = {W. Hodges},
DOI = {10.2307/2273551},
ZBLNUMBER = {0525.03024},
} -
[Po2] B. Poizat, Groupes Stables, Lyon: Bruno Poizat, 1987, vol. 2.
@book {Po2, MRKEY = {0902156},
AUTHOR = {Poizat, Bruno},
TITLE = {Groupes Stables},
SERIES = {Nur Al-Mantiq Wal-Ma'rifah [Light of Logic and Knowledge]},
VOLUME={2},
PUBLISHER = {Bruno Poizat},
ADDRESS = {Lyon},
YEAR = {1987},
PAGES = {vi+218},
ISBN = {2-9500919-1-1},
MRCLASS = {03C45 (03C60 12L12 20A15)},
MRNUMBER = {0902156},
MRREVIEWER = {B. I. Zil$'$ber},
ZBLNUMBER = {0633.03019},
} -
[Po3] B. Poizat, Stable Groups, Providence, RI: Amer. Math. Soc., 2001, vol. 87.
@book {Po3, MRKEY = {1827833},
AUTHOR = {Poizat, Bruno},
TITLE = {Stable Groups},
SERIES = {Math. Surveys Monogr.},
VOLUME = {87},
NOTE = {translated from the 1987 French original by Moses Gabriel Klein},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2001},
PAGES = {xiv+129},
ISBN = {0-8218-2685-9},
MRCLASS = {03C45 (03-02 03C60 20A15)},
MRNUMBER = {1827833},
ZBLNUMBER = {0969.03047},
} -
[Se1] Z. Sela, "Diophantine geometry over groups. I. Makanin-Razborov diagrams," Publ. Math. Inst. Hautes Études Sci., iss. 93, pp. 31-105, 2001.
@article {Se1, MRKEY = {1863735},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {I}. {M}akanin-{R}azborov diagrams},
JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
NUMBER = {93},
YEAR = {2001},
PAGES = {31--105},
ISSN = {0073-8301},
MRCLASS = {20F65 (20E05 20E18 20E26)},
MRNUMBER = {1863735},
MRREVIEWER = {Jos{é} Burillo},
DOI = {10.1007/s10240-001-8188-y},
ZBLNUMBER = {1018.20034},
} -
[Se2] Z. Sela, "Diophantine geometry over groups. II. Completions, closures and formal solutions," Israel J. Math., vol. 134, pp. 173-254, 2003.
@article {Se2, MRKEY = {1972179},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {II}. {C}ompletions, closures and formal solutions},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {134},
YEAR = {2003},
PAGES = {173--254},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {20F65 (03B25 20E05 20F10)},
MRNUMBER = {1972179},
MRREVIEWER = {Jos{é} Burillo},
DOI = {10.1007/BF02787407},
ZBLNUMBER = {1028.20028},
} -
[Se3] Z. Sela, "Diophantine geometry over groups. III. Rigid and solid solutions," Israel J. Math., vol. 147, pp. 1-73, 2005.
@article {Se3, MRKEY = {2166355},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {III}. {R}igid and solid solutions},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {147},
YEAR = {2005},
PAGES = {1--73},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {20F65 (03C10 20E05 20F10)},
MRNUMBER = {2166355},
MRREVIEWER = {Jos{é} Burillo},
DOI = {10.1007/BF02785359},
ZBLNUMBER = {1133.20020},
} -
[Se4] Z. Sela, "Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence," Israel J. Math., vol. 143, pp. 1-130, 2004.
@article {Se4, MRKEY = {2106978},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {IV}. {A}n iterative procedure for validation of a sentence},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {143},
YEAR = {2004},
PAGES = {1--130},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {20F65 (20E05 20F10)},
MRNUMBER = {2106978},
MRREVIEWER = {Jos{é} Burillo},
DOI = {10.1007/BF02803494},
ZBLNUMBER = {1088.20017},
} -
[Se5] Z. Sela, "Diophantine geometry over groups. $ V_1$. Quantifier elimination. I," Israel J. Math., vol. 150, pp. 1-197, 2005.
@article {Se5, MRKEY = {2249582},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {$\rm V\sb 1$}. {Q}uantifier elimination. {I}},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {150},
YEAR = {2005},
PAGES = {1--197},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {20F65 (03C10 20E05 20F10)},
MRNUMBER = {2249582},
MRREVIEWER = {Jos{é} Burillo},
DOI = {10.1007/BF02762378},
ZBLNUMBER = {1148.20022},
} -
[Se6] Z. Sela, "Diophantine geometry over groups. ${ V}_2$. Quantifier elimination. II," Geom. Funct. Anal., vol. 16, iss. 3, pp. 537-706, 2006.
@article {Se6, MRKEY = {2238944},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {${\rm V}\sb 2$}. {Q}uantifier elimination. {II}},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {16},
YEAR = {2006},
NUMBER = {3},
PAGES = {537--706},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {20F65 (03C10 20E05 20F10)},
MRNUMBER = {2238944},
MRREVIEWER = {Jos{é} Burillo},
ZBLNUMBER = {1118.20034},
DOI = {10.1007/s00039-006-0564-9},
} -
[Se7] Z. Sela, "Diophantine geometry over groups. VI. The elementary theory of a free group," Geom. Funct. Anal., vol. 16, iss. 3, pp. 707-730, 2006.
@article {Se7, MRKEY = {2238945},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {VI}. {T}he elementary theory of a free group},
JOURNAL = {Geom. Funct. Anal.},
FJOURNAL = {Geometric and Functional Analysis},
VOLUME = {16},
YEAR = {2006},
NUMBER = {3},
PAGES = {707--730},
ISSN = {1016-443X},
CODEN = {GFANFB},
MRCLASS = {20F65 (03C07 03C10 03C65 20E05 20F10)},
MRNUMBER = {2238945},
MRREVIEWER = {Daniel P. Groves},
DOI = {10.1007/s00039-006-0565-8},
} -
[Se8] Z. Sela, "Diophantine geometry over groups. VII. The elementary theory of a hyperbolic group," Proc. Lond. Math. Soc., vol. 99, iss. 1, pp. 217-273, 2009.
@article {Se8, MRKEY = {2520356},
AUTHOR={Sela, Z.},
TITLE = {Diophantine geometry over groups. {VII}. {T}he elementary theory of a hyperbolic group},
JOURNAL = {Proc. Lond. Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {99},
YEAR = {2009},
NUMBER = {1},
PAGES = {217--273},
ISSN = {0024-6115},
MRCLASS = {20F67 (03C10 20F70)},
MRNUMBER = {2520356},
MRREVIEWER = {Daniel P. Groves},
DOI = {10.1112/plms/pdn052},
ZBLNUMBER = {1241.20049},
} -
[Se9] Z. Sela, Diophantine geometry over groups IX: Envelopes and imaginaries.
@misc{Se9,
author={Sela, Z.},
TITLE={Diophantine geometry over groups {IX: E}nvelopes and imaginaries},
NOTE={preprint},
SORTYEAR={2012},
ARXIV = {0909.0774},
}
Full Article