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Diophantine geometry over groups. VIII: Stability. (English) Zbl 1285.20042

From the introduction: 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.
To prove the stability of free and hyperbolic groups, we start by analyzing a special class of definable sets that we call minimal rank. These sets are easier to analyze than general definable sets, and in Section 1 we prove that minimal rank definable sets are in the Boolean algebra generated by equational sets. (Recall that equational sets and theories were defined by G. Srour. For a definition, see the beginning of Section 1 and A. Pillay and G. Srour [J. Symb. Logic 49, 1350-1362 (1984; Zbl 0597.03018)].)
In Section 2 we slightly modify the sieve procedure that was presented in part V\(_2\) [Geom. Funct. Anal. 16, No. 3, 537-706 (2006; Zbl 1118.20034)] (and used for quantifier elimination) to prove that Diophantine sets are equational. The equationality of Diophantine sets is essentially equivalent to the termination of the sieve procedure for quantifier elimination in part V\(_2\) [loc. cit.], and it is a key in obtaining stability for general definable sets in the sequel. In Section 3 we present a basic object that we use repeatedly in proving stability – Duo ‘limit groups’ (Definition 3.1) – and their ‘rectangles’ (Definition 3.2). We further prove a boundedness property of duo limit groups and their rectangles (Theorem 3.3), which is not required in the sequel but still motivates our approach to stability.
In Section 4 we use duo limit groups and their rectangles, together with the sieve procedure and the equationality of Diophantine sets, to prove the stability of some families of definable sets, which are in a sense the building blocks of general definable sets (over a free group). These include the set of values of the defining parameters of a rigid and solid limit groups for which the rigid (solid) limit group has precisely \(s\) rigid (strictly solid families of) specializations for some fixed integer \(s\). (See [part I, Publ. Math., Inst. Hautes Étud. Sci. 93, 31-105 (2001; Zbl 1018.20034), §10] and [part III, Isr. J. Math. 147, 1-73 (2005; Zbl 1133.20020), §1] for these notions.)
In Section 5 we use the geometric structure of a general definable set that was proved using the sieve procedure in part V\(_2\) [loc. cit.], together with the stability of the families of definable sets that are considered in Section 4, to prove the stability of a general definable set over a free group and hence to obtain the stability of a free group (Theorem 5.1). Using the results of part VII [Proc. Lond. Math. Soc. (3) 99, No. 1, 217-273 (2009; Zbl 1241.20049)] we further generalize our results to a nonelementary, torsion-free (Gromov) hyperbolic group (Theorem 5.2).
The objects, techniques and arguments that we use in proving stability are all based on the work on Tarski’s problems and, in particular, on the sieve procedure for quantifier elimination ([part I-part V\(_2\)]). Parts of the arguments require not only familiarity with the main objects that are presented in these papers, but also with the procedures that are used in them. We give the exact references wherever we apply these procedures, or we use previously defined notions.

MSC:

20F70 Algebraic geometry over groups; equations over groups
20F67 Hyperbolic groups and nonpositively curved groups
20E05 Free nonabelian groups
03C45 Classification theory, stability, and related concepts in model theory
03B25 Decidability of theories and sets of sentences
03C07 Basic properties of first-order languages and structures
03C60 Model-theoretic algebra
20A15 Applications of logic to group theory
14A22 Noncommutative algebraic geometry
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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[1] R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, New York: Springer-Verlag, 1977, vol. 89. · Zbl 0368.20023
[2] A. Pillay, An Introduction to Stability Theory, New York: The Clarendon Press Oxford University Press, 1983, vol. 8. · Zbl 0526.03014
[3] A. Pillay and G. Srour, ”Closed sets and chain conditions in stable theories,” J. Symbolic Logic, vol. 49, iss. 4, pp. 1350-1362, 1984. · Zbl 0597.03018 · doi:10.2307/2274284
[4] B. Poizat, ”Groupes stables, avec types génériques réguliers,” J. Symbolic Logic, vol. 48, iss. 2, pp. 339-355, 1983. · Zbl 0525.03024 · doi:10.2307/2273551
[5] B. Poizat, Groupes Stables, Lyon: Bruno Poizat, 1987, vol. 2. · Zbl 0633.03019
[6] B. Poizat, Stable Groups, Providence, RI: Amer. Math. Soc., 2001, vol. 87. · Zbl 0969.03047
[7] Z. Sela, ”Diophantine geometry over groups. I. Makanin-Razborov diagrams,” Publ. Math. Inst. Hautes Études Sci., iss. 93, pp. 31-105, 2001. · Zbl 1018.20034 · doi:10.1007/s10240-001-8188-y
[8] Z. Sela, ”Diophantine geometry over groups. II. Completions, closures and formal solutions,” Israel J. Math., vol. 134, pp. 173-254, 2003. · Zbl 1028.20028 · doi:10.1007/BF02787407
[9] Z. Sela, ”Diophantine geometry over groups. III. Rigid and solid solutions,” Israel J. Math., vol. 147, pp. 1-73, 2005. · Zbl 1133.20020 · doi:10.1007/BF02785359
[10] Z. Sela, ”Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence,” Israel J. Math., vol. 143, pp. 1-130, 2004. · Zbl 1088.20017 · doi:10.1007/BF02803494
[11] Z. Sela, ”Diophantine geometry over groups. \( V_1\). Quantifier elimination. I,” Israel J. Math., vol. 150, pp. 1-197, 2005. · Zbl 1148.20022 · doi:10.1007/BF02762378
[12] Z. Sela, ”Diophantine geometry over groups. \({ V}_2\). Quantifier elimination. II,” Geom. Funct. Anal., vol. 16, iss. 3, pp. 537-706, 2006. · Zbl 1118.20034 · doi:10.1007/s00039-006-0564-9
[13] 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. · Zbl 1118.20035 · doi:10.1007/s00039-006-0565-8
[14] 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. · Zbl 1241.20049 · doi:10.1112/plms/pdn052
[15] Z. Sela, Diophantine geometry over groups IX: Envelopes and imaginaries.
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