Sela's early important work was his solution[1] in mid-1990s of the isomorphism problem for torsion-free word-hyperbolic groups. The machinery of group actions on real trees, developed by Eliyahu Rips, played a key role in Sela's approach. The solution of the isomorphism problem also relied on the notion of canonical representatives for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper.[12] The machinery of the canonical representatives allowed Rips and Sela to prove[12] algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in free groups, where the Makanin–Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani[13] to the case of relatively hyperbolic groups and played a key role in the solution of the isomorphism problem for toral relatively hyperbolic groups.[14]
In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups,[15] motivated by the notion of a JSJ decomposition for 3-manifolds. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclicsubgroups. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groups[16] and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians.[17][18][19][20] Sela applied a combination of his JSJ-decomposition and real tree techniques to prove that torsion-free word-hyperbolic groups are Hopfian.[21] This result and Sela's approach were later generalized by others to finitely generatedsubgroups of hyperbolic groups[22] and to the setting of relatively hyperbolic groups.
Sela's most important work came in early 2000s when he produced a solution to a famous Tarski conjecture. Namely, in a long series of papers,[23][24][25][26][27][28][29] he proved that any two non-abelian finitely generatedfree groups have the same first-order theory. Sela's work relied on applying his earlier JSJ-decomposition and real tree techniques as well as developing new ideas and machinery of "algebraic geometry" over free groups.
Sela pushed this work further to study first-order theory of arbitrary torsion-free word-hyperbolic groups and to characterize all groups that are elementarily equivalent to (that is, have the same first-order theory as) a given torsion-free word-hyperbolic group. In particular, his work implies that if a finitely generated group G is elementarily equivalent to a word-hyperbolic group then G is word-hyperbolic as well.
Sela also proved that the first-order theory of a finitely generated free group is stable in the model-theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory.
The work of Sela on first-order theory of free and word-hyperbolic groups substantially influenced the development of geometric group theory, in particular by stimulating the development and the study of the notion of limit groups and of relatively hyperbolic groups.[34]
Sela's classification theoremedit
Theorem. Two non-abelian torsion-free hyperbolic groups are elementarily equivalent if and only if their cores are isomorphic.[35]
Sela, Zlil (1995), "The isomorphism problem for hyperbolic groups", Annals of Mathematics, Second Series, 141 (2): 217–283, doi:10.2307/2118520, JSTOR 2118520, MR 1324134
Sela, Zlil (1997), "Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II.", Geometric and Functional Analysis, 7 (3): 561–593, doi:10.1007/s000390050019, MR 1466338, S2CID 120486267
Sela, Zlil; Rips, Eliyahu (1997), "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition", Annals of Mathematics, Second Series, 146 (1): 53–109, doi:10.2307/2951832, JSTOR 2951832, MR 1469317
Sela, Zlil (1997), "Acylindrical accessibility for groups", Inventiones Mathematicae, 129 (3): 527–565, Bibcode:1997InMat.129..527S, doi:10.1007/s002220050172, S2CID 122548154 (Sela's theorem on acylindrical accessibility for groups)[36]
Sela, Zlil (2001), "Diophantine geometry over groups. I. Makanin-Razborov diagrams" (PDF), Publications Mathématiques de l'IHÉS, 93 (1): 31–105, doi:10.1007/s10240-001-8188-y, MR 1863735, S2CID 51799226
Sela, Zlil (2003), "Diophantine geometry over groups. II. Completions, closures and formal solutions", Israel Journal of Mathematics, 134 (1): 173–254, doi:10.1007/BF02787407, MR 1972179
^ abZ. Sela. "The isomorphism problem for hyperbolic groups. I." Annals of Mathematics (2), vol. 141 (1995), no. 2, pp. 217–283.
^ abZ. Sela. Diophantine geometry over groups and the elementary theory of free and hyperbolic groups. Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87 92, Higher Ed. Press, Beijing, 2002. ISBN 7-04-008690-5
^ abFaculty Members Win Fellowships Columbia University Record, May 15, 1996, Vol. 21, No. 27.
^ abZ. Sela, and E. Rips. Canonical representatives and equations in hyperbolic groups, Inventiones Mathematicae vol. 120 (1995), no. 3, pp. 489–512
^François Dahmani. "Accidental parabolics and relatively hyperbolic groups."
Israel Journal of Mathematics, vol. 153 (2006), pp. 93–127
^François Dahmani, and Daniel Groves, "The isomorphism problem for toral relatively hyperbolic groups". Publications Mathématiques de l'IHÉS, vol. 107 (2008), pp. 211–290
^Z. Sela. "Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II." Geometric and Functional Analysis, vol. 7 (1997), no. 3, pp. 561–593
^E. Rips, and Z. Sela. "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition." Annals of Mathematics (2), vol. 146 (1997), no. 1, pp. 53–109
^M. J. Dunwoody, and M. E. Sageev. "JSJ-splittings for finitely presented groups over slender groups." Inventiones Mathematicae, vol. 135 (1999), no. 1, pp. 25 44
^P. Scott and G. A. Swarup. "Regular neighbourhoods and canonical decompositions for groups." Electronic Research Announcements of the American Mathematical Society, vol. 8 (2002), pp. 20–28
^B. H. Bowditch. "Cut points and canonical splittings of hyperbolic groups." Acta Mathematica, vol. 180 (1998), no. 2, pp. 145–186
^K. Fujiwara, and P. Papasoglu, "JSJ-decompositions of finitely presented groups and complexes of groups." Geometric and Functional Analysis, vol. 16 (2006), no. 1, pp. 70–125
^Sela, Z. (1999). "Endomorphisms of hyperbolic groups. I. The Hopf property". Topology. 38 (2): 301–321. doi:10.1016/S0040-9383(98)00015-9. MR 1660337.
^Inna Bumagina, "The Hopf property for subgroups of hyperbolic groups." Geometriae Dedicata, vol. 106 (2004), pp. 211–230
^Z. Sela. "Diophantine geometry over groups. I. Makanin-Razborov diagrams." Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 31–105
^Z. Sela. Diophantine geometry over groups. II. Completions, closures and formal solutions.Israel Journal of Mathematics, vol. 134 (2003), pp. 173–254
^Z. Sela. "Diophantine geometry over groups. III. Rigid and solid solutions." Israel Journal of Mathematics, vol. 147 (2005), pp. 1–73
^Z. Sela. "Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence." Israel Journal of Mathematics, vol. 143 (2004), pp. 1–130
^Z. Sela. "Diophantine geometry over groups. V1. Quantifier elimination. I." Israel Journal of Mathematics, vol. 150 (2005), pp. 1–197
^Z. Sela. "Diophantine geometry over groups. V2. Quantifier elimination. II." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 537–706
^Z. Sela. "Diophantine geometry over groups. VI. The elementary theory of a free group." Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 707–730
^O. Kharlampovich, and A. Myasnikov. "Tarski's problem about the elementary theory of free groups has a positive solution." Electronic Research Announcements of the American Mathematical Society, vol. 4 (1998), pp. 101–108
^O. Kharlampovich, and A. Myasnikov. Implicit function theorem over free groups. Journal of Algebra, vol. 290 (2005), no. 1, pp. 1–203
^O. Kharlampovich, and A. Myasnikov. "Algebraic geometry over free groups: lifting solutions into generic points." Groups, languages, algorithms, pp. 213–318, Contemporary Mathematics, vol. 378, American Mathematical Society, Providence, RI, 2005
^O. Kharlampovich, and A. Myasnikov. "Elementary theory of free non-abelian groups." Journal of Algebra, vol. 302 (2006), no. 2, pp. 451–552
^Frédéric Paulin.
Sur la théorie élémentaire des groupes libres (d'après Sela). Astérisque No. 294 (2004), pp. 63–402
^Guirardel, Vincent; Levitt, Gilbert; Salinos, Rizos (2020). "Towers and the first-order theory of hyperbolic groups". arXiv:2007.14148 [math.GR]. (See p. 8.)
^Kapovich, Ilya; Weidmann, Richard (2002). "Acylindrical accessibility for groups acting on R-tree". arXiv:math/0210308.