Brian Bowditch

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]

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]

• Bowditch, Brian H. (1995), "Geometrical finiteness with variable negative curvature", Duke Mathematical Journal, 77: 229–274, doi:10.1215/S0012-7094-95-07709-6, MR 1317633
• Bowditch, Brian H. (1998), "A topological characterisation of hyperbolic groups", Journal of the American Mathematical Society, 11 (3): 643–667, doi:10.1090/S0894-0347-98-00264-1, MR 1602069
• Bowditch, Brian H. (1998), "Cut points and canonical splittings of hyperbolic groups", Acta Mathematica, 180 (2): 145–186, doi:10.1007/BF02392898, MR 1638764
• Bowditch, Brian H. (2006), "Intersection numbers and the hyperbolicity of the curve complex", Crelle's Journal, 2006 (598): 105–129, doi:10.1515/CRELLE.2006.070, MR 2270568, S2CID 10831464
• Bowditch, Brian H. (2007), "The angel game in the plane", Combinatorics, Probability and Computing, 16 (3): 345–362, doi:10.1017/S0963548306008297, MR 2312431
• Bowditch, Brian H. (2008), "Tight geodesics in the curve complex", Inventiones Mathematicae, 171 (2): 281–300, doi:10.1007/s00222-007-0081-y, MR 2367021, S2CID 18808639

• Geometric group theory
• Geometric topology
• 3-manifolds
• Kleinian groups

• Brian H. Bowditch's HomePage at the University of Warwick