Brian Bowditch

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Brian Hayward Bowditch (born 1961[1]) is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving[2] the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.

Biography[edit]

Brian Bowditch was born in 1961 in Neath, Wales. He obtained a B.A. degree from Cambridge University in 1983.[1] He subsequently pursued doctoral studies in Mathematics at the University of Warwick under the supervision of David Epstein where he received a PhD in 1988.[3] Bowditch then had postdoctoral and visiting positions at the Institute for Advanced Study in Princeton, the University of Warwick, Institut des Hautes Études Scientifiques at Bures-sur-Yvette, the University of Melbourne, and the University of Aberdeen.[1] In 1992 he received an appointment at the University of Southampton where he stayed until 2007. In 2007 Bowditch moved to the University of Warwick, where he receive a chaired Professor appointment in Mathematics.

Bowditch was awarded a Whitehead Prize by the London Mathematical Society in 2007 for his work in geometric group theory and geometric topology.[4][5]

Bowditch gave an Invited address at the 2004 European Congress of Mathematics in Stockholm.[6]

Brian Bowditch is a member of the Editorial Board for the journal Annales de la Faculté des Sciences de Toulouse[7] and a former Editorial Adviser for the London Mathematical Society.[8]

Mathematical contributions[edit]

Early notable results of Bowditch include clarifying the classic notion of geometric finiteness for higher-dimensional Kleinian groups in constant and variable negative curvature. In a 1993 paper[9] Bowditch proved that five standard characterisations of geometric finiteness for discrete groups of isometries of hyperbolic 3-space and hyperbolic plane, (including the definition in terms of having a finitely-sided fundamental polyhedron) remain equivalent for groups of isometries of hyperbolic n-space where n ≥ 4. He showed, however, that in dimensions n ≥ 4 the condition of having a finitely-sided Dirichlet domain is no longer equivalent to the standard notions of geometric finiteness. In a subsequent paper[10] Bowditch considered a similar problem for discrete groups of isometries of Hadamard manifold of pinched (but not necessarily constant) negative curvature and of arbitrary dimension n ≥ 2. He proved that four out of five equivalent definitions of geometric finiteness considered in his previous paper remain equivalent in this general set-up, but the condition of having a finitely-sided fundamental polyhedron is no longer equivalent to them.

Much of Bowditch's work in the 1990s concerned studying boundaries at infinity of word-hyperbolic groups. He proved the cut-point conjecture which says that the boundary of a one-ended word-hyperbolic group does not have any global cut-points. Bowditch first proved this conjecture in the main cases of a one-ended hyperbolic group that does not split over a two-ended subgroup[11] (that is, a subgroup containing infinite cyclic subgroup of finite index) and also for one-ended hyperbolic groups that are "strongly accessible".[12] The general case of the conjecture was finished shortly thereafter by Swarup[13] who characterised Bowditch's work as follows: "The most significant advances in this direction were carried out by Brian Bowditch in a brilliant series of papers ([4]-[7]). We draw heavily from his work". Soon after Swarup's paper Bowditch supplied an alternative proof of the cut-point conjecture in the general case.[14] Bowditch's work relied on extracting various discrete tree-like structures from the action of a word-hyperbolic group on its boundary.

Bowditch also proved that (modulo a few exceptions) the boundary of a one-ended word-hyperbolic group G has local cut-points if and only if G admits an essential splitting, as an amalgamated free product or an HNN extension, over a virtually infinite cyclic group. This allowed Bowditch to produce[15] a theory of JSJ-decomposition for word-hyperbolic groups that was more canonical and more general (particularly because it covered groups with nontrivial torsion) than the original JSJ-decomposition theory of Zlil Sela.[16] One of the consequences of Bowditch's work is that for one-ended word-hyperbolic groups (with a few exceptions) having a nontrivial essential splitting over a virtually cyclic subgroup is a quasi-isometry invariant.

Bowditch also gave a topological characterisation of word-hyperbolic groups, thus solving a conjecture proposed by Mikhail Gromov. Namely, Bowditch proved[17] that a group G is word-hyperbolic if and only if G admits an action by homeomorphisms on a perfect metrisable compactum M as a "uniform convergence group", that is such that the diagonal action of G on the set of distinct triples from M is properly discontinuous and co-compact; moreover, in that case M is G-equivariantly homeomorphic to the boundary ∂G of G. Later, building up on this work, Bowditch's PhD student Yaman gave a topological characterisation of relatively hyperbolic groups.[18]

Much of Bowditch's work in 2000s concerns the study of the curve complex, with various applications to 3-manifolds, mapping class groups and Kleinian groups. The curve complex C(S) of a finite type surface S, introduced by Harvey in the late 1970s,[19] has the set of free homotopy classes of essential simple closed curves on S as the set of vertices, where several distinct vertices span a simplex if the corresponding curves can be realised disjointly. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space, of mapping class groups and of Kleinian groups. In a 1999 paper[20] Masur and Minsky proved that for a finite type orientable surface S the curve complex C(S) is Gromov-hyperbolic. This result was a key component in the subsequent proof of Thurston's Ending lamination conjecture, a solution which was based on the combined work of Minsky, Masur, Brock and Canary.[21] In 2006 Bowditch gave another proof[22] of hyperbolicity of the curve complex. Bowditch's proof is more combinatorial and rather different from the Masur-Minsky original argument. Bowditch's result also provides an estimate on the hyperbolicity constant of the curve complex which is logarithmic in complexity of the surface and also gives a description of geodesics in the curve complex in terms of the intersection numbers. A subsequent 2008 paper of Bowditch[23] pushed these ideas further and obtained new quantitative finiteness results regarding the so-called "tight geodesics" in the curve complex, a notion introduced by Masur and Minsky to combat the fact that the curve complex is not locally finite. As an application, Bowditch proved that, with a few exceptions of surfaces of small complexity, the action of the mapping class group Mod(S) on C(S) is "acylindrical" and that the asymptotic translation lengths of pseudo-anosov elements of Mod(S) on C(S) are rational numbers with bounded denominators.

A 2007 paper of Bowditch[2] produces a positive solution of the angel problem of John Conway:[24] Bowditch proved[2] that a 4-angel has a winning strategy and can evade the devil in the "angel game". Independent solutions of the angel problem were produced at about the same time by Máthé[25] and Kloster.[26]

Selected publications[edit]

See also[edit]

References[edit]

  1. ^ a b c Brian H. Bowditch: Me. Bowditch's personal information page at the University of Warwick
  2. ^ a b c B. H. Bowditch, "The angel game in the plane" Combinatorics, Probability and Computing, vol. 16 (2007), no. 3, pp. 345–362
  3. ^ Brian Hayward Bowditch at the Mathematics Genealogy Project
  4. ^ Lynne Williams. "Awards" Times Higher Education, 24 October 1997
  5. ^ "Records of Proceedings at Meetings" Bulletin of the London Mathematical Society, vol 30 (1998), pp. 438–448; Quote from the Whitehead Prize award citation for Brian Bowditch, pp. 445–446:"Bowditch has made significant and totally original contributions to hyperbolic geometry, especially to the associated group theory. [...] His deepest work is on the asymptotic properties of word-hyperbolic groups. This work simultaneously generalises and simplifies recent work of several authors, and it already has many applications. In one application, he develops a new theory of groups acting on dendrites. Building on previous contributions of G. Levitt, G. A. Swarup and others, this led him to a solution of the 'cut-point conjecture'. This recent work also yields a characterisation of word-hyperbolic groups as convergence groups. Bowditch has solved several major problems in geometric group theory using methods that are elegant and as elementary as they can be."
  6. ^ European Congress of Mathematics, Stockholm, June 27 – July 2, 2004 European Mathematical Society, 2005. ISBN 978-3-03719-009-8
  7. ^ Editorial Board, Annales de la Faculté des Sciences de Toulouse. Accessed 15 October 2008
  8. ^ London Mathematical Society 2005 publications London Mathematical Society. Accessed 15 October 2008.
  9. ^ B. H. Bowditch, "Geometrical finiteness for hyperbolic groups" Journal of Functional Analysis, vol. 113 (1993), no. 2, 245–317
  10. ^ B. H. Bowditch, "Geometrical finiteness with variable negative curvature" Duke Mathematical Journal, vol. 77 (1995), no. 1, 229–274
  11. ^ B. H. Bowditch, "Group actions on trees and dendrons" Topology, vol. 37 (1998), no. 6, pp. 1275–1298
  12. ^ B. H. Bowditch, "Boundaries of strongly accessible hyperbolic groups" The Epstein birthday schrift, pp. 51–97, Geometry&Topology Monographs, vol. 1, Geom. Topol. Publ., Coventry, 1998
  13. ^ G. A. Swarup, "On the cut point conjecture" Electronic Research Announcements of the American Mathematical Society, vol. 2 (1996), no. 2, pp. 98–100
  14. ^ B. H. Bowditch, "Connectedness properties of limit sets" Transactions of the American Mathematical Society, vol. 351 (1999), no. 9, pp. 3673–3686
  15. ^ B. H. Bowditch, "Cut points and canonical splittings of hyperbolic groups" Acta Mathematica, vol. 180 (1998), no. 2, 145–186.
  16. ^ 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
  17. ^ B. H. Bowditch, "A topological characterisation of hyperbolic groups" Journal of the American Mathematical Society, vol. 11 (1998), no. 3, pp. 643–667.
  18. ^ Asli Yaman. "A topological characterisation of relatively hyperbolic groups". Crelle's Journal, vol. 566 (2004), pp. 41–89.
  19. ^ W. J. Harvey, "Boundary structure of the modular group". Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), pp. 245–251, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981. ISBN 0-691-08264-2
  20. ^ Howard Masur, and Yair Minsky, "Geometry of the complex of curves. I. Hyperbolicity" Inventiones Mathematicae, vol. 138 (1999), no. 1, pp. 103–149.
  21. ^ Yair Minsky, "Curve complexes, surfaces and 3-manifolds". International Congress of Mathematics. Vol. II, pp. 1001–1033, Eur. Math. Soc., Zürich, 2006. ISBN 978-3-03719-022-7
  22. ^ Brian H. Bowditch, "Intersection numbers and the hyperbolicity of the curve complex" Crelle's Journal, vol. 598 (2006), pp. 105–129.
  23. ^ Brian H. Bowditch, "Tight geodesics in the curve complex" Inventiones Mathematicae, vol. 171 (2008), no. 2, pp. 281–300.
  24. ^ John H. Conway, "The angel problem" Games of no chance (Berkeley, California, 1994), pp. 3–12, Mathematical Sciences Research Institute Publications, 29, Cambridge University Press, Cambridge, 1996. ISBN 0-521-57411-0
  25. ^ A. Máthé, "The angel of power 2 wins" Combinatorics, Probability and Computing, vol. 16 (2007), no. 3, pp. 363–374
  26. ^ O. Kloster, "A solution to the angel problem" Theoretical Computer Science, vol. 389 (2007), no. 1-2, pp. 152–161

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