Brian Bowditch

Last updated

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.

Contents

Biography

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, New Jersey, 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 received a chaired Professor appointment in Mathematics.

Bowditch was awarded a Whitehead Prize by the London Mathematical Society in 1997 for his work in geometric group theory and geometric topology. [4] [5] He gave an Invited address at the 2004 European Congress of Mathematics in Stockholm. [6] Bowditch is a former 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

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 G. Ananda 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] Howard Masur and Yair 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 Yair Minsky, Howard Masur, Jeffrey Brock, and Richard 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 András Máthé [25] and Oddvar Kloster. [26]

Selected publications

See also

Related Research Articles

<span class="mw-page-title-main">Discrete group</span>

In mathematics, a topological group G is called a discrete group if there is no limit point in it. Equivalently, the group G is discrete if and only if its identity is isolated.

<span class="mw-page-title-main">Low-dimensional topology</span> Branch of topology

In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

<span class="mw-page-title-main">Geometric group theory</span> Area in mathematics devoted to the study of finitely generated groups

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.

<span class="mw-page-title-main">3-manifold</span> Mathematical space

In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.

<span class="mw-page-title-main">Kleinian group</span> Discrete group of Möbius transformations

In mathematics, a Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of hyperbolic 3-space H3. The latter, identifiable with PSL(2, C), is the quotient group of the 2 by 2 complex matrices of determinant 1 by their center, which consists of the identity matrix and its product by −1. PSL(2, C) has a natural representation as orientation-preserving conformal transformations of the Riemann sphere, and as orientation-preserving conformal transformations of the open unit ball B3 in R3. The group of Möbius transformations is also related as the non-orientation-preserving isometry group of H3, PGL(2, C). So, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

<span class="mw-page-title-main">Hyperbolic group</span> Mathematical concept

In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or Gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. The notion of a hyperbolic group was introduced and developed by Mikhail Gromov (1987). The inspiration came from various existing mathematical theories: hyperbolic geometry but also low-dimensional topology, and combinatorial group theory. In a very influential chapter from 1987, Gromov proposed a wide-ranging research program. Ideas and foundational material in the theory of hyperbolic groups also stem from the work of George Mostow, William Thurston, James W. Cannon, Eliyahu Rips, and many others.

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold.

James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.

<span class="mw-page-title-main">Mladen Bestvina</span> Croatian-American mathematician

Mladen Bestvina is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.

In hyperbolic geometry, the ending lamination theorem, originally conjectured by William Thurston (1982), states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.

In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.

In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by Namazi & Souto (2012) and Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.

<span class="mw-page-title-main">Yair Minsky</span>

Yair Nathan Minsky is an Israeli-American mathematician whose research concerns three-dimensional topology, differential geometry, group theory and holomorphic dynamics. He is a professor at Yale University. He is known for having proved Thurston's ending lamination conjecture and as a student of curve complex geometry.

In mathematics, the curve complex is a simplicial complex C(S) associated to a finite-type surface S, which encodes the combinatorics of simple closed curves on S. 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. It was introduced by W.J.Harvey in 1978.

In mathematics, a convergence group or a discrete convergence group is a group acting by homeomorphisms on a compact metrizable space in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary of the hyperbolic 3-space . The notion of a convergence group was introduced by Gehring and Martin (1987) and has since found wide applications in geometric topology, quasiconformal analysis, and geometric group theory.

<span class="mw-page-title-main">Jeffrey Brock</span> American mathematician

Jeffrey Farlowe Brock is an American mathematician, working in low-dimensional geometry and topology. He is known for his contributions to the understanding of hyperbolic 3-manifolds and the geometry of Teichmüller spaces.

Albert Marden is an American mathematician, specializing in complex analysis and hyperbolic geometry.

In mathematics, a Cannon–Thurston map is any of a number of continuous group-equivariant maps between the boundaries of two hyperbolic metric spaces extending a discrete isometric actions of the group on those spaces.

References

  1. 1 2 3 Brian H. Bowditch: Me. Bowditch's personal information page at the University of Warwick
  2. 1 2 3 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 Gilbert Levitt, G. Ananda 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 Archived 17 July 2011 at the Wayback Machine 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 Archived 27 October 2005 at the Wayback Machine London Mathematical Society. Accessed 15 October 2008.
  9. ↑ Bowditch, B.H. (1993), "Geometrical Finiteness for Hyperbolic Groups" (PDF), Journal of Functional Analysis, 113 (2): 245–317, doi:10.1006/jfan.1993.1052
  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. ↑ Zlil 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" [ permanent dead link ] 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. ↑ András Máthé, "The angel of power 2 wins" Combinatorics, Probability and Computing , vol. 16 (2007), no. 3, pp. 363–374 MR 2312432
  26. ↑ Oddvar Kloster, "A solution to the angel problem" Theoretical Computer Science , vol. 389 (2007), no. 1-2, pp. 152–161 MR 2363369
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.