Zlil Sela

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Zlil Sela

Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution[1] of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first order theories of finitely generated non-abelian free groups.[2]

Biographical data[edit]

Sela received his Ph.D. in 1991 from the Hebrew University of Jerusalem, where his doctoral advisor was Eliyahu Rips. Prior to his current appointment at the Hebrew University, he held an Associate Professor position at Columbia University in New York.[3] While at Columbia, Sela won the Sloan Fellowship from the Sloan Foundation.[3][4]

Sela gave an Invited Address at the 2002 International Congress of Mathematicians in Beijing.[2][5] He gave a plenary talk at the 2002 annual meeting of the Association for Symbolic Logic,[6] and he delivered an AMS Invited Address at the October 2003 meeting of the American Mathematical Society[7] and the 2005 Tarski Lectures (other languages) at the University of California at Berkeley.[8] He was also awarded the 2003 Erdős Prize from the Israel Mathematical Union.[9] Sela also received the 2008 Carol Karp Prize from the Association for Symbolic Logic for his work on the Tarski conjecture and on discovering and developing new connections between model theory and geometric group theory.[10][11]

Mathematical contributions[edit]

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 hyprbolic 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 cyclic subgroups. 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 generated subgroups 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 generated free 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.

An alternative solution for the Tarski conjecture has been presented by Olga Kharlampovich and Alexei Myasnikov.[30][31][32][33]

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]

Published work[edit]

See also[edit]

References[edit]

  1. ^ a b Z. Sela. The isomorphism problem for hyperbolic groups. I. Annals of Mathematics (2), vol. 141 (1995), no. 2, pp. 217–283.
  2. ^ a b Z. 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
  3. ^ a b Faculty Members Win Fellowships Columbia University Record, May 15, 1996, Vol. 21, No. 27.
  4. ^ Sloan Fellowships Awarded Notices of the American Mathematical Society, vol. 43 (1996), no. 7, pp. 781–782
  5. ^ Invited Speakers for ICM2002. Notices of the American Mathematical Society, vol. 48, no. 11, December 2001; pp. 1343 1345
  6. ^ The 2002 annual meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic, vol. 9 (2003), pp. 51–70
  7. ^ AMS Meeting at Binghamton, New York. Notices of the American Mathematical Society, vol. 50 (2003), no. 9, p. 1174
  8. ^ 2005 Tarski Lectures. Department of Mathematics, University of California at Berkeley. Accessed September 14, 2008.
  9. ^ Erdős Prize. Israel Mathematical Union. Accessed September 14, 2008
  10. ^ Karp Prize Recipients. Association for Symbolic Logic. Accessed September 13, 2008
  11. ^ ASL Karp and Sacks Prizes Awarded, Notices of the American Mathematical Society, vol. 56 (2009), no. 5, p. 638
  12. ^ a b Z. Sela, and E. Rips. Canonical representatives and equations in hyperbolic groups, Inventiones Mathematicae vol. 120 (1995), no. 3, pp. 489–512
  13. ^ François Dahmani. Accidental parabolics and relatively hyperbolic groups. Israel Journal of Mathematics, vol. 153 (2006), pp. 93–127
  14. ^ François Dahmani, and Daniel Groves, The isomorphism problem for toral relatively hyperbolic groups. Publications Mathématiques de l'IHÉS, vol. 107 (2008), pp. 211–290
  15. ^ 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
  16. ^ 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
  17. ^ 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
  18. ^ 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
  19. ^ B. H. Bowditch. Cut points and canonical splittings of hyperbolic groups. Acta Mathematica, vol. 180 (1998), no. 2, pp. 145–186
  20. ^ 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
  21. ^ Zlil Sela, Endomorphisms of hyperbolic groups. I. The Hopf property. Topology, vol. 38 (1999), no. 2, pp. 301–321
  22. ^ Inna Bumagina, The Hopf property for subgroups of hyperbolic groups. Geometriae Dedicata, vol. 106 (2004), pp. 211–230
  23. ^ Z. Sela. Diophantine geometry over groups. I. Makanin-Razborov diagrams. Publications Mathématiques. Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 31–105
  24. ^ Z. Sela. Diophantine geometry over groups. II. Completions, closures and formal solutions. Israel Journal of Mathematics, vol. 134 (2003), pp. 173–254
  25. ^ Z. Sela. Diophantine geometry over groups. III. Rigid and solid solutions. Israel Journal of Mathematics, vol. 147 (2005), pp. 1–73
  26. ^ 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
  27. ^ Z. Sela. Diophantine geometry over groups. V1. Quantifier elimination. I. Israel Journal of Mathematics, vol. 150 (2005), pp. 1–197
  28. ^ Z. Sela. Diophantine geometry over groups. V2. Quantifier elimination. II. Geometric and Functional Analysis, vol. 16 (2006), no. 3, pp. 537–706
  29. ^ 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
  30. ^ 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
  31. ^ O. Kharlampovich, and A. Myasnikov. Implicit function theorem over free groups. Journal of Algebra, vol. 290 (2005), no. 1, pp. 1–203
  32. ^ 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
  33. ^ O. Kharlampovich, and A. Myasnikov. Elementary theory of free non-abelian groups. Journal of Algebra, vol. 302 (2006), no. 2, pp. 451–552
  34. ^ Frédéric Paulin. Sur la théorie élémentaire des groupes libres (d'après Sela). Astérisque No. 294 (2004), pp. 63–402

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