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'''John Robert Stallings''' (1935&ndash;2008) was a mathematician known for his seminal contributions to [[geometric group theory]] and [[Low-dimensional topology|3-manifold topology]]. Stallings was a Professor Emeritus in the Department of Mathematics at the [[University of California at Berkeley]].<ref>[http://math.berkeley.edu/index.php?module=mathfacultyman&MATHFACULTY_MAN_op=sView&MATHFACULTY_id=142 UC Berkeley Department of Mathematics faculty profile.]</ref>
'''John Robert Stallings''' (1935&ndash;2008) was a mathematician known for his seminal contributions to [[geometric group theory]] and [[Low-dimensional topology|3-manifold topology]]. Stallings was a Professor Emeritus in the Department of Mathematics at the [[University of California at Berkeley]].<ref>[http://math.berkeley.edu/index.php?module=mathfacultyman&MATHFACULTY_MAN_op=sView&MATHFACULTY_id=142 UC Berkeley Department of Mathematics faculty profile.]</ref>
Stallings received his B.Sc. from [[University of Arkansas]] in 1956 (where he was one of the first two graduates in the university's Honors program)<ref>[http://libinfo.uark.edu/ata/v3no4/honorscollege.asp All things academic.] Volume 3, Issue 4; November 2002. </ref> and he received a Ph.D. in Mathematics from [[Princeton University]] in 1959 under the direction of [[Ralph Fox]]. Stallings had 22 doctoral students and 60 doctoral descendants. He published over 50 papers, predominantly in the areas of [[geometric group theory]] and the topology of [[3-manifold]]s.
Stallings received his B.Sc. from [[University of Arkansas]] in 1956 (where he was one of the first two graduates in the university's Honors program)<ref>[http://libinfo.uark.edu/ata/v3no4/honorscollege.asp All things academic.] Volume 3, Issue 4; November 2002. </ref> and he received a Ph.D. in Mathematics from [[Princeton University]] in 1959 under the direction of [[Ralph Fox]]. Stallings had 22 doctoral students and 60 doctoral descendants. He published over 50 papers, predominantly in the areas of [[geometric group theory]] and the topology of [[3-manifold]]s.

Revision as of 14:04, 2 December 2008

John Robert Stallings (1935–2008) was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley.[1] Stallings received his B.Sc. from University of Arkansas in 1956 (where he was one of the first two graduates in the university's Honors program)[2] and he received a Ph.D. in Mathematics from Princeton University in 1959 under the direction of Ralph Fox. Stallings had 22 doctoral students and 60 doctoral descendants. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds.

Stallings delivered an invited address as the International Congress of Mathematicians in Nice in 1970[3] and a James K. Whittemore Lecture at Yale University in 1969.[4]

Stallings received the Frank Nelson Cole Prize in Algebra from the American Mathematical Society in 1970.[5]

The conference "Geometric and Topological Aspects of Group Theory", held at the Mathematical Sciences Research Institute in Berkeley in May 2000, was dedicated to the 65th birthday of Stallings. [6] In 2002 a special issue of the journal Geometriae Dedicata was dedicated to Stallings on the occasion of his 65th birthday.[7]

Mathematical contributions

Most of Stallings' mathematical contributions are in the areas of geometric group theory and low-dimensional topology (particularly the topology of 3-manifolds) and on the interplay between these two areas.

An early significant result of Stallings is his 1960 proof[8] of the analog of the Poincaré Conjecture in dimensions greater than six. (Stallings' proof was obtained independently from and at about the same time as the proof of Steve Smale who established the same result in dimensions bigger than four[9]).

Using methods similar to those in his proof of the Poincaré Conjecture for n > 6, Stallings proved that ordinary Euclidean n-dimensional space has a unique piecewise linear, hence also smooth, structure, if n is not equal to 4. This took on added significance when, as a consequence of work of Michael Freedman and Simon Donaldson in 1982, it was shown that 4-space has exotic smooth structures, in fact uncountably many such.

Stallings' most famous theorem is an algebraic characterization of groups with more than one end (that is, with more than one "connected component at infinity"), that is now known as Stallings theorem about ends of groups. Stallings proved that a finitely generated group G has more than one end if and only if this group admits a nontrivial splitting as an amalgamated free product or as an HNN-extension over a finite group (that is, in terms of Bass-Serre theory, if and only if the group admits a nontrivial action on a tree with finite edge stabilizers). More precisely, the theorem states that a finitely generated group G has more than one end if and only if either G admits a splitting as an amalgamated free product , where the group C is finite and C≠A, C≠B, or G admits a splitting as an HNN-extension where K,L≤H are finite subgroups of H.

Stallings proved this result in a series of works, first dealing with the torsion-free case (that is a group with no nontrivial elements of finite order)[10] and then with the general case.[11][4] Stalling's theorem yielded a positive solution to the long-standing open problem about characterizing finitely generated groups of cohomological dimension one as exactly the free groups.[12]. Stallings' theorem about ends of groups is considered one of the first results in geometric group theory proper since it connects a geometric property of a group (having more than one end) with its algebraic structure (admitting a splitting over a finite subgroup). Stallings' theorem spawned many subsequent alternative proofs by other mathematicians (e.g. [13][14]) as well as many applications (e.g. [15]). The theorem also motivated several generalizations and relative versions of Stallings' result to other contexts, such as the study of the notion of relative ends of a group with respect to a subgroup,[16][17][18] including a connection to CAT(0) cubical complexes.[19]. A comprehensive survey discussing, in particular, numerous applications and generalizations of Stallings' theorem, is given in a 2003 paper of Wall.[20]

Another influential paper of Stalling is his 1983 article "Topology on finite graphs".[21] Traditionally, the algebraic structure of subgroups of free groups has been studied in combinatorial group theory using combinatorial methods, such as the Schreier rewriting method and Nielsen transformations.[22] Stallings' paper put forward a topological approach based on the methods of covering space theory that also used a simple graph-theoretic framework. The paper introduced the notion of what is now commonly referred to as Stallings subgroup graph for describing subgroups of free groups, and also introduced a foldings technique (used for approximating and algorithmically obtaining the subgroup graphs) and the notion of what is now known as a Stallings folding. Most classical results regarding subgroups of free groups acquired simple and straightforward proofs in this set-up and Stallings' method has become the standard tool in the theory for studying the subgroup structure of free groups, including both the algebraic and algorithmic questions (see [23]). In particular, Stallings subgroup graphs and Stallings foldings have been the used as a key tools in many attempts to approach the Hanna Neumann Conjecture.[24][25][26][27]

Stallings subgroup graphs can also be viewed as finite state automata[23] and they have also found applications in semigroup theory and in computer science.[28][29][30][31]

Stallings' foldings method has been generalized and applied to other contexts, particularly in Bass-Serre theory for approximating group actions on trees and studying the subgroup structure of the fundamental groups of graphs of groups. The first paper in this direction was written by Stallings himself[32], with several subsequent generalizations of Stallings' folding methods in the Bass-Serre theory context by other mathematicians. [33][34][35] [36]

Stallings' 1991 paper "Non-positively curved triangles of groups"[37]introduced and studied the notion of a triangle of groups. This notion was the starting point for the theory of complexes of groups (a higher-dimensional analog of Bass-Serre theory), developed by Haefliger[38] and others. [39][40] Stallings' work pointed out the importance of imposing some sort of "non-positive curvature" conditions on the complexes of groups in order for the theory to work well; such restrictions are not necessary in the one-dimensional case of Bass-Serre theory.

Among Stallings' contributions to low-dimensional topology, the most well-known is Stallings' fibration theorem.[41] The theorem states that if M is a compact irreducible 3-manifold whose fundamental group contains a normal subgroup, such that this subgroup is finitely generated and such that the quotient group by this subgroup is infinite cyclic, then M fibers over a circle. This is an important structural result in the theory of Haken manifolds that engendered many alternative proofs, generalizations and applications (e.g. [42][43][44][45] ), including a higher-dimensional analog[46].

A 1965 paper of Stallings "How not to prove the Poincaré conjecture"[47] gave a group-theoretic reformulation of the famous Poincaré conjecture. Despite its ironic title, Stallings' paper informed much of the subsequent research on exploring the algebraic aspects of the Poincaré Conjecture (see, for example, [48][49][50][51]).

Selected works

  • Stallings, John R. (1960), "Polyhedral homotopy spheres", Bulletin of the American Mathematical Society, 66: 485–488, MR0124905 {{citation}}: External link in |title= (help)
  • Stallings, John R. (1962), "The piecewise-linear structure of Euclidean space", Proceedings of the Cambridge Philosophical Society, 58: 481–488, MR0149457
  • Stallings, John R. (1962), "On fibering certain 3-manifolds", Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice Hall, pp. 95–100, MR0158375
  • Stallings, John R. (1965), "Homology and central series of groups", Journal of Algebra, 2: 170–181, doi:10.1016/0021-8693(65)90017-7, ISSN 0021-8693, MR0175956
  • Stallings, John R. (1968), "On torsion-free groups with infinitely many ends", Annals of Mathematics. Second Series, 88: 312–334, doi:10.2307/1970577, ISSN 0003-486X, MR0228573 {{citation}}: External link in |title= (help)
  • Stallings, John R. (1971), Group theory and three-dimensional manifolds, Yale University Press, MR0415622
  • Stallings, John R. (1978), "Constructions of fibred knots and links", Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, Proc. Sympos. Pure Math., XXXII, Providence, R.I.: American Mathematical Society, pp. 55–60, MR520522
  • Stallings, John R. (1983), "Topology of finite graphs", Inventiones Mathematicae, 71 (3): 551–565, doi:10.1007/BF02095993, ISSN 0020-9910, MR695906 {{citation}}: External link in |title= (help), with over 100 recent citations
  • Stallings, John R. (1991), "Folding G-trees", Arboreal group theory (Berkeley, CA, 1988), Mathematical Sciences Research Institute Publications, vol. 19, New York: Springer, pp. 355–368, ISBN 0-387-97518-7, MR1105341
  • Stallings, John R. (1991), "Non-positively curved triangles of groups", Group theory from a geometrical viewpoint (Trieste, 1990), River Edge, NJ: World Scientific, pp. 491–903, ISBN 981-02-0442-6, MR1170374

References

  1. ^ UC Berkeley Department of Mathematics faculty profile.
  2. ^ All things academic. Volume 3, Issue 4; November 2002.
  3. ^ John R. Stallings. Group theory and 3-manifolds. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 165–167. Gauthier-Villars, Paris, 1971.
  4. ^ a b John Stallings. Group theory and three-dimensional manifolds. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.–London, 1971.
  5. ^ Frank Nelson Cole Prize in Algebra. American Mathematical Society.
  6. ^ Geometric and Topological Aspects of Group Theory, conference announcement, atlas-conferences.com
  7. ^ Geometriae Dedicata, vol. 92 (2002). Special issue dedicated to John Stallings on the occasion of his 65th birthday. Edited by R. Z. Zimmer.
  8. ^ John Stallings. Polyhedral homotopy spheres. Bulletin of the American Mathematical Society, vol. 66 (1960), pp. 485–488.
  9. ^ S. Smale. Generalized Poincaré's conjecture in dimensions greater than four. Annals of Mathematics (2nd Ser.), vol. 74 (1961), no. 2, pp. 391–406
  10. ^ John R. Stallings. On torsion-free groups with infinitely many ends. Annals of Mathematics (2), vol. 88 (1968), pp. 312–334.
  11. ^ John Stallings. Groups of cohomological dimension one. Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVIII, New York, 1968) pp. 124–128. American Mathematical Society, Providence, R.I, 1970.
  12. ^ John R. Stallings. Groups of dimension 1 are locally free. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364
  13. ^ M. J.Dunwoody. Cutting up graphs. Combinatorica 2 (1982), no. 1, pp. 15–23.
  14. ^ Warren Dicks, and M. J. Dunwoody. Groups acting on graphs. Cambridge Studies in Advanced Mathematics, 17. Cambridge University Press, Cambridge, 1989. ISBN 0-521-23033-0
  15. ^ Peter Scott. A new proof of the annulus and torus theorems. American Journal of Mathematics, vol. 102 (1980), no. 2, pp. 241–277
  16. ^ G. A.Swarup. Relative version of a theorem of Stallings. Journal of Pure and Applied Algebra, vol. 11 (1977/78), no. 1–3, pp. 75–82
  17. ^ M. J. Dunwoody, and E. L. Swenson. The algebraic torus theorem. Inventiones Mathematicae, vol. 140 (2000), no. 3, pp. 605–637
  18. ^ G. P. Scott, and G. A. Swarup. An algebraic annulus theorem. Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506
  19. ^ Michah Sageev. Ends of group pairs and non-positively curved cube complexes. Proceedings of the London Mathematical Society (3), vol. 71 (1995), no. 3, pp. 585–617
  20. ^ C. T. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16 (2003), no. 1, pp. 5–101.
  21. ^ John R. Stallings. Topology of finite graphs. Inventiones Mathematicae, vol. 71 (1983), no. 3, pp. 551–565
  22. ^ Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer–Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ISBN-13: 9783540411581
  23. ^ a b Ilya Kapovich, and Alexei Myasnikov. Stallings foldings and subgroups of free groups. Journal of Algebra, vol. 248 (2002), no. 2, 608–668
  24. ^ J. Meakin, and P. Weil. Subgroups of free groups: a contribution to the Hanna Neumann conjecture. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000). Geometriae Dedicata, vol. 94 (2002), pp. 33–43.
  25. ^ Warren Dicks. Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture. Inventiones Mathematicae, vol. 117 (1994), no. 3, pp. 373–389.
  26. ^ Warren Dicks, and Edward Formanek. The rank three case of the Hanna Neumann conjecture. Journal of Group Theory, vol. 4 (2001), no. 2, pp. 113–151
  27. ^ Bilal Khan. Positively generated subgroups of free groups and the Hanna Neumann conjecture. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 155–170, Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002; ISBN 0-8218-2822-3
  28. ^ Jean-Camille Birget, and Stuart W. Margolis. Two-letter group codes that preserve aperiodicity of inverse finite automata. Semigroup Forum, vol. 76 (2008), no. 1, pp. 159–168
  29. ^ D. S. Ananichev, A. Cherubini, M. V. Volkov. Image reducing words and subgroups of free groups. Theoretical Computer Science, vol. 307 (2003), no. 1, pp. 77–92.
  30. ^ J. Almeida, and M. V. Volkov. Subword complexity of profinite words and subgroups of free profinite semigroups. International Journal of Algebra and Computation, vol. 16 (2006), no. 2, pp. 221–258.
  31. ^ Benjamin Steinberg. A topological approach to inverse and regular semigroups. Pacific Journal of Mathematics, vol. 208 (2003), no. 2, pp. 367–396
  32. ^ John R. Stallings. Foldings of G-trees. Arboreal group theory (Berkeley, CA, 1988), pp. 355–368, Math. Sci. Res. Inst. Publ., 19, Springer, New York, 1991; ISBN 0-387-97518-7
  33. ^ Mladen Bestvina and Mark Feighn. Bounding the complexity of simplicial group actions on trees, Inventiones Mathematicae, vol. 103, (1991), no. 3, pp. 449–469
  34. ^ M. J. Dunwoody. Folding sequences. The Epstein birthday schrift, pp. 139–158 (electronic), Geometry and Topology Monographs, 1, Geom. Topol. Publ., Coventry, 1998.
  35. ^ Ilya Kapovich, Richard Weidmann, and Alexei Miasnikov. Foldings, graphs of groups and the membership problem. International Journal of Algebra and Computation, vol. 15 (2005), no. 1, pp. 95–128.
  36. ^ Yuri Gurevich, and Paul E. Schupp. Membership problem for the modular group. SIAM Journal on Computing, vol. 37 (2007), no. 2, pp. 425–459
  37. ^ John R. Stallings. Non-positively curved triangles of groups. Group theory from a geometrical viewpoint (Trieste, 1990), pp. 491–503, World Sci. Publ., River Edge, NJ, 1991; ISBN 981-02-0442-6
  38. ^ André Haefliger. Complexes of groups and orbihedra. in: "Group theory from a geometrical viewpoint (Trieste, 1990)", pp. 504–540, World Sci. Publ., River Edge, NJ, 1991. ISBN 981-02-0442-6
  39. ^ Jon Corson. Complexes of groups. Proceedings of the London Mathematical Society (3) 65 (1992), no. 1, pp. 199–224.
  40. ^ Martin R. Bridson, and André Haefliger. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Springer-Verlag, Berlin, 1999. ISBN 3-540-64324-9
  41. ^ John R. Stallings. On fibering certain 3-manifolds. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 95–100. Prentice-Hall, Englewood Cliffs, N.J
  42. ^ John Hempel, and William Jaco. 3-manifolds which fiber over a surface. American Journal of Mathematics, vol. 94 (1972), pp. 189–205
  43. ^ Alois Scharf. Zur Faserung von Graphenmannigfaltigkeiten. (in German) Mathematische Annalen, vol. 215 (1975), pp. 35–45.
  44. ^ Louis Zulli. Semibundle decompositions of 3-manifolds and the twisted cofundamental group. Topology and its Applications, vol. 79 (1997), no. 2, pp. 159–172
  45. ^ Nathan M. Dunfield, and Dylan P. Thurston. A random tunnel number one 3-manifold does not fiber over the circle. Geometry & Topology, vol. 10 (2006), pp. 2431–2499
  46. ^ W. Browder, and J. Levine. Fibering manifolds over a circle. Commentarii Mathematici Helvetici, vol. 40 (1966), pp. 153–160
  47. ^ John R. Stallings. How not to prove the Poincaré conjecture. Topology Seminar, Wisconsin, 1965. Edited by R. H. Bing and R. J. Bean. Annals of Mathematics Studies, No. 60. Princeton University Press, Princeton, N.J. 1966
  48. ^ Robert Myers. Splitting homomorphisms and the geometrization conjecture. Mathematical Proceedings of the Cambridge Philosophical Society, vol. 129 (2000), no. 2, pp. 291–300
  49. ^ Tullio Ceccherini-Silberstein. On the Grigorchuk-Kurchanov conjecture. Manuscripta Mathematica 107 (2002), no. 4, pp. 451–461
  50. ^ V. N. Berestovskii. Poincaré's conjecture and related statements. (in Russian) Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. vol. 51 (2000), no. 9, pp. 3–41; translation in Russian Mathematics (Izvestiya VUZ. Matematika), vol. 51 (2007), no. 9, 1–36
  51. ^ V. Poenaru. Autour de l'hypothèse de Poincaré. in: "Géométrie au XXe siècle, 1930–2000 : histoire et horizons". Montréal, Presses internationales Polytechnique, 2005. ISBN 255301399X, 9782553013997.
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