31:
350:
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
266:
for elements of hyperbolic groups, introduced by Rips and Sela in a joint 1995 paper. The machinery of the canonical representatives allowed Rips and Sela to prove algorithmic solvability of finite systems of equations in torsion-free hyperbolic groups, by reducing the problem to solving equations in
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and this work gave rise a systematic development of the JSJ-decomposition theory with many further extensions and generalizations by other mathematicians. Sela applied a combination of his JSJ-decomposition and
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In his work on the isomorphism problem Sela also introduced and developed the notion of a JSJ-decomposition for word-hyperbolic groups, motivated by the notion of a
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271:, where the Makanin–Razborov algorithm can be applied. The technique of canonical representatives was later generalized by Dahmani to the case of
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Proceedings of the
International Congress of Mathematicians, Vol. II (Beijing, 2002), pp. 87 92, Higher Ed. Press, Beijing, 2002.
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1319:
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Theorem. Two non-abelian torsion-free hyperbolic groups are elementarily equivalent if and only if their cores are isomorphic.
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in the model-theoretic sense, providing a brand-new and qualitatively different source of examples for the stability theory.
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Guirardel, Vincent; Levitt, Gilbert; Salinos, Rizos (2020). "Towers and the first-order theory of hyperbolic groups".
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The work of Sela on first-order theory of free and word-hyperbolic groups substantially influenced the development of
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O. Kharlampovich, and A. Myasnikov. "Algebraic geometry over free groups: lifting solutions into generic points."
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507:(1997), "Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II.",
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E. Rips, and Z. Sela. "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition."
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142:
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Kapovich, Ilya; Weidmann, Richard (2002). "Acylindrical accessibility for groups acting on R-tree".
1333:
1096:
1048:"Structure and rigidity in (Gromov) hyperbolic groups and discrete groups in rank 1 Lie groups. II."
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techniques as well as developing new ideas and machinery of "algebraic geometry" over free groups.
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for his work on the Tarski conjecture and on discovering and developing new connections between
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552:(1997), "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition",
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172:
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Solution of the isomorphism problem for torsion-free word-hyperbolic groups, Tarski conjecture
1456:
1258:"Diophantine geometry over groups. IV. An iterative procedure for validation of a sentence."
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680:(2003), "Diophantine geometry over groups. II. Completions, closures and formal solutions",
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Diophantine geometry over groups and the elementary theory of free and hyperbolic groups.
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715:(2006), "Diophantine geometry over groups. VI. The elementary theory of a free group",
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Sela's most important work came in early 2000s when he produced a solution to a famous
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1334:"Tarski's problem about the elementary theory of free groups has a positive solution."
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O. Kharlampovich, and A. Myasnikov. "Elementary theory of free non-abelian groups."
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305:. The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free
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Sela also proved that the first-order theory of a finitely generated free group is
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Diophantine geometry over groups. II. Completions, closures and formal solutions.
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1316:"Diophantine geometry over groups. VI. The elementary theory of a free group."
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1013:
François
Dahmani. "Accidental parabolics and relatively hyperbolic groups."
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344:
311:
290:. A JSJ-decomposition is a representation of a word-hyperbolic group as the
255:
1131:"JSJ-decompositions of finitely presented groups and complexes of groups."
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332:. Namely, in a long series of papers, he proved that any two non-abelian
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of hyperbolic groups and to the setting of relatively hyperbolic groups.
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417:(1995), "Canonical representatives and equations in hyperbolic groups",
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An alternative solution for the Tarski conjecture has been presented by
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Electronic
Research Announcements of the American Mathematical Society
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Electronic
Research Announcements of the American Mathematical Society
318:. This result and Sela's approach were later generalized by others to
30:
1433:
1213:. Institut de Hautes Études Scientifiques, vol. 93 (2001), pp. 31–105
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and played a key role in the solution of the isomorphism problem for
194:
in
Beijing. He gave a plenary talk at the 2002 annual meeting of the
567:
481:
1411:
1242:"Diophantine geometry over groups. III. Rigid and solid solutions."
1079:"JSJ-splittings for finitely presented groups over slender groups."
343:. Sela's work relied on applying his earlier JSJ-decomposition and
1451:
1208:"Diophantine geometry over groups. I. Makanin-Razborov diagrams."
1097:"Regular neighbourhoods and canonical decompositions for groups."
1030:"The isomorphism problem for toral relatively hyperbolic groups".
314:
techniques to prove that torsion-free word-hyperbolic groups are
242:
Sela's early important work was his solution in mid-1990s of the
634:"Diophantine geometry over groups. I. Makanin-Razborov diagrams"
897:
The 2002 annual meeting of the
Association for Symbolic Logic.
1393:
Sur la théorie élémentaire des groupes libres (d'après Sela).
811:
Z. Sela. "The isomorphism problem for hyperbolic groups. I."
62:
Sloan
Fellowship, Erdős Prize (2003), Carol Karp Prize (2008)
998:
Canonical representatives and equations in hyperbolic groups
590:
Sela, Zlil (1997), "Acylindrical accessibility for groups",
1113:"Cut points and canonical splittings of hyperbolic groups."
966:
Association for
Symbolic Logic. Accessed September 13, 2008
355:
is elementarily equivalent to a word-hyperbolic group then
1150:"Endomorphisms of hyperbolic groups. I. The Hopf property"
855:
Columbia
University Record, May 15, 1996, Vol. 21, No. 27.
625:(Sela's theorem on acylindrical accessibility for groups)
466:(1995), "The isomorphism problem for hyperbolic groups",
1190:"The Hopf property for subgroups of hyperbolic groups."
947:
1352:
Journal of Algebra, vol. 290 (2005), no. 1, pp. 1–203
887:, vol. 48, no. 11, December 2001; pp. 1343 1345
1492:
Academic staff of the Hebrew University of Jerusalem
1365:, pp. 213–318, Contemporary Mathematics, vol. 378,
95:
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21:
1457:Zlil Sela at the Mathematics Genealogy Project
294:which encodes in a canonical way all possible
179:in New York. While at Columbia, Sela won the
175:, he held an Associate Professor position at
8:
1452:Zlil Sela's webpage at the Hebrew University
981:Notices of the American Mathematical Society
916:Notices of the American Mathematical Society
885:Notices of the American Mathematical Society
869:Notices of the American Mathematical Society
1350:Implicit function theorem over free groups.
171:. Prior to his current appointment at the
122:. He is a Professor of Mathematics at the
18:
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817:(2), vol. 141 (1995), no. 2, pp. 217–283.
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190:Sela gave an Invited Address at the 2002
163:Sela received his Ph.D. in 1991 from the
1497:Einstein Institute of Mathematics alumni
1086:, vol. 135 (1999), no. 1, pp. 25 44
192:International Congress of Mathematicians
126:. Sela is known for the solution of the
1068:(2), vol. 146 (1997), no. 1, pp. 53–109
801:
992:
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292:fundamental group of a graph of groups
1395:Astérisque No. 294 (2004), pp. 63–402
1382:, vol. 302 (2006), no. 2, pp. 451–552
1120:, vol. 180 (1998), no. 2, pp. 145–186
1028:François Dahmani, and Daniel Groves,
825:
823:
7:
1348:O. Kharlampovich, and A. Myasnikov.
1332:O. Kharlampovich, and A. Myasnikov.
1323:, vol. 16 (2006), no. 3, pp. 707–730
1305:, vol. 16 (2006), no. 3, pp. 537–706
1294:"Diophantine geometry over groups. V
1274:"Diophantine geometry over groups. V
1034:Publications Mathématiques de l'IHÉS
933:University of California at Berkeley
912:AMS Meeting at Binghamton, New York.
872:, vol. 43 (1996), no. 7, pp. 781–782
848:
846:
807:
805:
642:Publications Mathématiques de l'IHÉS
208:University of California at Berkeley
1477:21st-century Israeli mathematicians
1472:20th-century Israeli mathematicians
1138:, vol. 16 (2006), no. 1, pp. 70–125
1055:, vol. 7 (1997), no. 3, pp. 561–593
1004:vol. 120 (1995), no. 3, pp. 489–512
1077:M. J. Dunwoody, and M. E. Sageev.
976:ASL Karp and Sacks Prizes Awarded
14:
1320:Geometric and Functional Analysis
1302:Geometric and Functional Analysis
1135:Geometric and Functional Analysis
1052:Geometric and Functional Analysis
718:Geometric and Functional Analysis
510:Geometric and Functional Analysis
167:, where his doctoral advisor was
919:, vol. 50 (2003), no. 9, p. 1174
29:
1507:Israel Defense Prize recipients
1129:K. Fujiwara, and P. Papasoglu,
984:, vol. 56 (2009), no. 5, p. 638
853:Faculty Members Win Fellowships
210:. He was also awarded the 2003
1298:. Quantifier elimination. II."
1231:, vol. 134 (2003), pp. 173–254
1197:, vol. 106 (2004), pp. 211–230
1037:, vol. 107 (2008), pp. 211–290
935:. Accessed September 14, 2008.
351:if a finitely generated group
279:relatively hyperbolic groups.
224:Association for Symbolic Logic
218:. Sela also received the 2008
196:Association for Symbolic Logic
165:Hebrew University of Jerusalem
124:Hebrew University of Jerusalem
86:Hebrew University of Jerusalem
45:Hebrew University of Jerusalem
1:
1367:American Mathematical Society
1363:Groups, languages, algorithms
1281:Israel Journal of Mathematics
1278:. Quantifier elimination. I."
1261:Israel Journal of Mathematics
1245:Israel Journal of Mathematics
1229:Israel Journal of Mathematics
1167:10.1016/S0040-9383(98)00015-9
1019:, vol. 153 (2006), pp. 93–127
1016:Israel Journal of Mathematics
882:Invited Speakers for ICM2002.
683:Israel Journal of Mathematics
396:Sela's classification theorem
200:American Mathematical Society
1339:, vol. 4 (1998), pp. 101–108
1283:, vol. 150 (2005), pp. 1–197
1263:, vol. 143 (2004), pp. 1–130
390:relatively hyperbolic groups
359:is word-hyperbolic as well.
273:relatively hyperbolic groups
137:and for the solution of the
1247:, vol. 147 (2005), pp. 1–73
1095:P. Scott and G. A. Swarup.
931:Department of Mathematics,
1523:
1211:Publications Mathématiques
1102:, vol. 8 (2002), pp. 20–28
902:, vol. 9 (2003), pp. 51–70
900:Bulletin of Symbolic Logic
238:Mathematical contributions
865:Sloan Fellowships Awarded
780:Group isomorphism problem
731:10.1007/s00039-006-0565-8
655:10.1007/s10240-001-8188-y
307:finitely presented groups
264:canonical representatives
216:Israel Mathematical Union
105:
66:
28:
1083:Inventiones Mathematicae
1002:Inventiones Mathematicae
593:Inventiones Mathematicae
420:Inventiones Mathematicae
118:working in the area of
1502:Erdős Prize recipients
1369:, Providence, RI, 2005
996:Z. Sela, and E. Rips.
957:Karp Prize Recipients.
760:Geometric group theory
382:geometric group theory
248:word-hyperbolic groups
232:geometric group theory
135:word-hyperbolic groups
120:geometric group theory
1066:Annals of Mathematics
929:2005 Tarski Lectures.
814:Annals of Mathematics
775:Word-hyperbolic group
614:10.1007/s002220050172
555:Annals of Mathematics
523:10.1007/s000390050019
469:Annals of Mathematics
141:about equivalence of
16:Israeli mathematician
143:first-order theories
1194:Geometriae Dedicata
606:1997InMat.129..527S
433:1995InMat.120..489R
250:. The machinery of
244:isomorphism problem
177:Columbia University
128:isomorphism problem
90:Columbia University
1380:Journal of Algebra
962:2008-05-13 at the
697:10.1007/BF02787407
441:10.1007/BF01241140
371:Olga Kharlampovich
341:first-order theory
334:finitely generated
320:finitely generated
147:finitely generated
1391:Frédéric Paulin.
1148:Sela, Z. (1999).
790:JSJ decomposition
558:, Second Series,
472:, Second Series,
330:Tarski conjecture
284:JSJ decomposition
246:for torsion-free
173:Hebrew University
159:Biographical data
139:Tarski conjecture
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68:Scientific career
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101:Eliyahu Rips
82:Institutions
67:
1417:(See p. 8.)
337:free groups
288:3-manifolds
269:free groups
212:Erdős Prize
153:free groups
150:non-abelian
77:Mathematics
1466:Categories
1412:2007.14148
796:References
785:Real trees
770:Free group
713:Sela, Zlil
678:Sela, Zlil
630:Sela, Zlil
546:Sela, Zlil
505:Sela, Zlil
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411:Sela, Zlil
296:splittings
256:real trees
1314:Z. Sela.
1292:Z. Sela.
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1240:Z. Sela.
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747:123197664
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345:real tree
323:subgroups
312:real tree
303:subgroups
222:from the
214:from the
183:from the
112:Zlil Sela
35:Zlil Sela
23:Zlil Sela
1154:Topology
960:Archived
754:See also
671:51799226
632:(2001),
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