John R. Stallings

John Robert Stallings is a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings is a Professor Emeritus in the Department of Mathematics and the University of California at Berkeley. [ [http://math.berkeley.edu/index.php?module=mathfacultyman&MATHFACULTY_MAN_op=sView&MATHFACULTY_id=142 UC Berkeley Deppartment of Mathematics faculty profile.] ] 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) [ [http://libinfo.uark.edu/ata/v3no4/honorscollege.asp All things academic.] Volume 3, Issue 4; November 2002. ] and he received a Ph.D. in Mathematics from Princeton University in 1959 under the direction of Ralph Fox. Stallings has 22 doctoral students and 60 doctoral descendants. He has 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 [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.] and a James K. Whittemore Lecture at Yale University in 1969. 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.]

Stallings received the Frank Nelson Cole Prize in Algebra from the American Mathematical Society in 1970. [ [http://www.ams.org/prizes/cole-prize-algebra.html Frank Nelson Cole Prize in Algebra.] American Mathematical Society.]

The conference "Geometric and Topological Aspects of Group Theory", held at the Mathematical Sciences Research Institute in Berekely in May 2000, was dedicated to the 65th birthday of Stallings. [ [http://atlas-conferences.com/cgi-bin/calendar/d/faam71 Geometric and Topological Aspects of Group Theory, conference announcement] , atlas-conferences.com] In 2002 a special issue of the journal Geometriae Dedicata was dedicated to Stallings on the occasion of his 65th birthday. [ [http://www.springerlink.com/content/acnlhf5dylu1/?p=36fc0e096ab34a99bf226a2b5cd5ca0a&pi=0 Geometriae Dedicata,] vol. 92 (2002). Special issue dedicated to John Stallings on the occasion of his 65th birthday. Edited by R. Z. Zimmer.]

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 [John Stallings. "Polyhedral homotopy spheres." Bulletin of the American Mathematical Society, vol. 66 (1960), pp. 485–488.] of the analog of the Poincare 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 [S. Smale. "Generalized Poincaré's conjecture in dimensions greater than four". Annals of Mathematics (2nd Ser.), vol. 74 (1961), no. 2, pp. 391–406] ).

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 $scriptstyle\; G=Aast\_C\; B$, where the group "C" is finite and "C"≠"A", "C"≠"B", or "G" admits a splitting as an HNN-extension $scriptstyle\; G=langle\; H,\; t\; |\; t^\{-1\}Kt=L\; angle$ 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) [John R. Stallings. "On torsion-free groups with infinitely many ends." Annals of Mathematics (2), vol. 88 (1968), pp. 312–334.] and then with the general case. [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. ] 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. [John R. Stallings. "Groups of dimension 1 are locally free." Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364] . 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. [M. J.Dunwoody. "Cutting up graphs." Combinatorica 2 (1982), no. 1, pp. 15–23.] [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 ] ) as well as many applications (e.g. [Peter Scott. "A new proof of the annulus and torus theorems." American Journal of Mathematics, vol. 102 (1980), no. 2, pp. 241–277 ] ). 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, [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 ] [M. J. Dunwoody, and E. L. Swenson. "The algebraic torus theorem." Inventiones Mathematicae, vol. 140 (2000), no. 3, pp. 605–637 ] [G. P. Scott, and G. A. Swarup. "An algebraic annulus theorem." Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506 ] including a connection to CAT(0) cubical complexes. [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 ] . A comprehensive survey discussing, in particular, numerous applications and generalizations of Stallings' theorem, is given in a 2003 paper of Wall. [C. T. Wall. "The geometry of abstract groups and their splittings."Revista Matemática Complutense vol. 16 (2003), no. 1, pp. 5–101.]

Another influential paper of Stalling is his 1984 article "Topology on finite graphs". [John R. Stallings. [http://www.springerlink.com/content/mn2h645qw2058530/ "Topology of finite graphs."] Inventiones Mathematicae, vol. 71 (1983), no. 3, pp. 551–565] 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. [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] 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 of a Stallings fold. 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 Ilya Kapovich, and Alexei Myasnikov. "Stallings foldings and subgroups of free groups." Journal of Algebra, vol. 248 (2002), no. 2, 608–668] ). 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. [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.] [Warren Dicks. "Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture." Inventiones Mathematicae, vol. 117 (1994), no. 3, pp. 373–389.] [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] [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]

Stallings subgroup graphs can also be viewed as finite state automata and they have also found applications in semigroup theory and in computer science. [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] [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.] [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.] [Benjamin Steinberg. "A topological approach to inverse and regular semigroups." Pacific Journal of Mathematics, vol. 208 (2003), no. 2, pp. 367–396]

Stallings' foldings method was also generalized and applied to other context, 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 [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] , with several subsequent generalizations of Stallings' folding methods in the Bass-Serre theory context by other mathematicians. [Mladen Bestvina and Mark Feign. "Bounding the complexity of simplicial group actions on trees", Inventiones Mathematicae, vol. 103, (1991), no. 3, pp. 449–469] [M. J. Dunwoody. [http://msp.warwick.ac.uk/gtm/1998/01/p007.xhtml "Folding sequences."] The Epstein birthday schrift, pp. 139–158 (electronic),Geometry and Topology Monographs, 1, Geom. Topol. Publ., Coventry, 1998.] [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.] [Yuri Gurevich, and Paul E. Schupp. "Membership problem for the modular group." SIAM Journal on Computing, vol. 37 (2007), no. 2, pp. 425–459]

Stallings' 1991 paper "Non-positively curved triangles of groups" [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] 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 [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] and others. [Jon Corson. "Complexes of groups." Proceedings of the London Mathematical Society (3) 65 (1992), no. 1, pp. 199–224.] [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] 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. [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 ] 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 produced many alternative proofs, generalizations and applications (e.g. [John Hempel, and William Jaco. "3-manifolds which fiber over a surface." American Journal of Mathematics, vol. 94 (1972), pp. 189–205] [Alois Scharf. "Zur Faserung von Graphenmannigfaltigkeiten." (in German)Mathematische Annalen, vol. 215 (1975), pp. 35–45.] [Louis Zulli. "Semibundle decompositions of 3-manifolds and the twisted cofundamental group." Topology and its Applications, vol. 79 (1997), no. 2, pp. 159–172 ] [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 ] ), including a higher-dimensional analog [W. Browder, and J. Levine."Fibering manifolds over a circle." Commentarii Mathematici Helvetici, vol. 40 (1966), pp. 153–160 ] .

A 1965 paper of Stallings "How not to prove the Poincaré conjecture" [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] 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, [Robert Myers. "Splitting homomorphisms and the geometrization conjecture." Mathematical Proceedings of the Cambridge Philosophical Society, vol. 129 (2000), no. 2, pp. 291–300 ] [Tullio Ceccherini-Silberstein. "On the Grigorchuk-Kurchanov conjecture." Manuscripta Mathematica 107 (2002), no. 4, pp. 451–461 ] [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 ] [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.] ).

Selected works

* | year=1960 | journal=Bulletin of the American Mathematical Society | volume=66 | pages=485–488
* | year=1962 | journal=Proceedings of the Cambridge Philosophical Society | volume=58 | pages=481–488
*Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) | publisher=Prentice Hall | id=MathSciNet | id = 0158375 | year=1962 | chapter=On fibering certain 3-manifolds | pages=95–100
* | year=1965 | journal=Journal of Algebra | issn=0021-8693 | volume=2 | pages=170–181
* | year=1968 | journal=Annals of Mathematics. Second Series | issn=0003-486X | volume=88 | pages=312–334
*Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Group theory and three-dimensional manifolds | publisher=Yale University Press | id=MathSciNet | id = 0415622 | year=1971
*Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2 | publisher=American Mathematical Society | location=Providence, R.I. | series=Proc. Sympos. Pure Math., XXXII | id=MathSciNet | id = 520522 | year=1978 | chapter=Constructions of fibred knots and links | pages=55–60
* | year=1983 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=71 | issue=3 | pages=551–565, with over 100 recent citations
*Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Arboreal group theory (Berkeley, CA, 1988)| publisher=Springer | location=New York | series=Mathematical Sciences Research Institute Publications|volume =19 | id=MathSciNet | id =1105341 | year=1991 | chapter=Folding "G"-trees| pages=355–368| isbn=0-387-97518-7
*Citation | last1=Stallings | first1=John R. | author1-link=John R. Stallings | title=Group theory from a geometrical viewpoint (Trieste, 1990) | publisher=World Scientific | location=River Edge, NJ| id=MathSciNet | id =1170374 | year=1991 | chapter=Non-positively curved triangles of groups | pages=491–903|isbn=981-02-0442-6

References

External links

*
* [http://math.berkeley.edu/~stall/ home page] of John Stallings.