Brian Bowditch
Encyclopedia
Brian Hayward Bowditch 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 the angel problem
. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick
.
, Wales
. He obtained a B.A. degree from Cambridge University in 1983. 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. 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
. 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
.
Bowditch gave an Invited address at the 2004 European Congress of Mathematicians in Stockholm.
Brian Bowditch is a member of the Editorial Board for the journal Annales de la Faculté des Sciences de Toulouse and a former Editorial Adviser for the London Mathematical Society
.
s in constant and variable negative curvature. In a 1993 paper Bowditch proved that five standard characterizations 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 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-point
s. Bowditch first proved this conjecture in the main cases of a one-ended hyperbolic group that does not split over a two-ended subgroup
(that is, a subgroup containing infinite cyclic subgroup of finite index
) and also for one-ended hyperbolic groups that are "strongly accessible". The general case of the conjecture was finished shortly thereafter by Swarup who characterized 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. 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 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
. 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 characterization of word-hyperbolic groups, thus solving a conjecture proposed by Mikhail Gromov. Namely, Bowditch proved that a group G is word-hyperbolic if and only if G admits an action
by homeomorphism
s on a perfect metrizable 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 characterization of relatively hyperbolic group
s.
Much of Bowditch's work in 2000s concerns the study of the curve complex, with various applications to 3-manifold
s, mapping class group
s and Kleinian group
s. The curve complex C(S) of a finite type surface S, introduced by Harvey in the late 1970s, 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 realized disjointly. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space
, of mapping class group
s and of Kleinian group
s. In a 1999 paper 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. In 2006 Bowditch gave another proof 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 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 produces a positive solution of the angel problem
of John Conway
: Bowditch proved 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 and Kloster.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
known for his contributions to geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, particularly in the areas of geometric group theory
Geometric group theory
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 .Another important...
and low-dimensional topology
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
. He is also known for solving the angel problem
Angel problem
The angel problem is a question in game theory proposed by John Horton Conway. The game is commonly referred to as the Angels and Devils game. The game is played by two players called the angel and the devil. It is played on an infinite chessboard...
. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick
University of Warwick
The University of Warwick is a public research university located in Coventry, United Kingdom...
.
Biography
Brian Bowditch was born in 1961 in NeathNeath
Neath is a town and community situated in the principal area of Neath Port Talbot, Wales, UK with a population of approximately 45,898 in 2001...
, Wales
Wales
Wales is a country that is part of the United Kingdom and the island of Great Britain, bordered by England to its east and the Atlantic Ocean and Irish Sea to its west. It has a population of three million, and a total area of 20,779 km²...
. He obtained a B.A. degree from Cambridge University in 1983. He subsequently pursued doctoral studies in Mathematics at the University of Warwick
University of Warwick
The University of Warwick is a public research university located in Coventry, United Kingdom...
under the supervision of David Epstein
David Epstein
David Epstein may refer to:* David Epstein , writer at Sports Illustrated* David B. A. Epstein , British mathematician* David G. Epstein, professor at the University of Richmond School of Law and bankruptcy expertSee also:...
where he received a PhD in 1988. Bowditch then had postdoctoral and visiting positions at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
in Princeton
Princeton, New Jersey
Princeton is a community located in Mercer County, New Jersey, United States. It is best known as the location of Princeton University, which has been sited in the community since 1756...
, the University of Warwick
University of Warwick
The University of Warwick is a public research university located in Coventry, United Kingdom...
, Institut des Hautes Études Scientifiques
Institut des Hautes Études Scientifiques
The Institut des Hautes Études Scientifiques is a French institute supporting advanced research in mathematics and theoretical physics...
at Bures-sur-Yvette, the University of Melbourne
University of Melbourne
The University of Melbourne is a public university located in Melbourne, Victoria. Founded in 1853, it is the second oldest university in Australia and the oldest in Victoria...
, and the University of Aberdeen
University of Aberdeen
The University of Aberdeen, an ancient university founded in 1495, in Aberdeen, Scotland, is a British university. It is the third oldest university in Scotland, and the fifth oldest in the United Kingdom and wider English-speaking world...
. In 1992 he received an appointment at the University of Southampton
University of Southampton
The University of Southampton is a British public university located in the city of Southampton, England, a member of the Russell Group. The origins of the university can be dated back to the founding of the Hartley Institution in 1862 by Henry Robertson Hartley. In 1902, the Institution developed...
where he stayed until 2007. In 2007 Bowditch moved to the University of Warwick
University of Warwick
The University of Warwick is a public research university located in Coventry, United Kingdom...
, where he receive a chaired Professor appointment in Mathematics.
Bowditch was awarded a Whitehead Prize
Whitehead Prize
The Whitehead Prize is awarded yearly by the London Mathematical Society to a mathematician working in the United Kingdom who is at an early stage of their career. The prize is named in memory of homotopy theory pioneer J. H. C...
by the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
in 2007 for his work in geometric group theory
Geometric group theory
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 .Another important...
and geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
.
Bowditch gave an Invited address at the 2004 European Congress of Mathematicians in Stockholm.
Brian Bowditch is a member of the Editorial Board for the journal Annales de la Faculté des Sciences de Toulouse and a former Editorial Adviser for the London Mathematical Society
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
.
Mathematical contributions
Early notable results of Bowditch include clarifying the classic notion of geometric finiteness for higher-dimensional Kleinian groupKleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s in constant and variable negative curvature. In a 1993 paper Bowditch proved that five standard characterizations of geometric finiteness for discrete groups of isometries of hyperbolic 3-space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
and hyperbolic plane
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, (including the definition in terms of having a finitely-sided fundamental polyhedron) remain equivalent for groups of isometries of hyperbolic n-space
Hyperbolic space
In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...
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 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
Stallings theorem about ends of groups
In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup...
word-hyperbolic group does not have any global cut-point
Cut-point
In topology, a cut-point is a point of a connected space such that its removal causes the resulting space to be disconnected. For example every point of a line is a cut-point, while no point of a circle is a cut-point...
s. Bowditch first proved this conjecture in the main cases of a one-ended hyperbolic group that does not split over a two-ended subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
(that is, a subgroup containing infinite cyclic subgroup of finite index
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
) and also for one-ended hyperbolic groups that are "strongly accessible". The general case of the conjecture was finished shortly thereafter by Swarup who characterized 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. Bowditch's work relied on extracting various discrete tree-like structures from the action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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 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
Zlil Sela
Zlil Sela is an Israeli mathematician working in the area of geometric group theory.He is a Professor of Mathematics at the Hebrew University of Jerusalem...
. 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
Quasi-isometry
In mathematics, a quasi-isometry is a means to compare the large-scale structure of metric spaces. The concept is especially important in Gromov's geometric group theory.-Definition:...
invariant.
Bowditch also gave a topological characterization of word-hyperbolic groups, thus solving a conjecture proposed by Mikhail Gromov. Namely, Bowditch proved that a group G is word-hyperbolic if and only if G admits an action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
by homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s on a perfect metrizable 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 characterization of relatively hyperbolic group
Relatively hyperbolic group
In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group...
s.
Much of Bowditch's work in 2000s concerns the study of the curve complex, with various applications to 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s, mapping class group
Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
s and Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s. The curve complex C(S) of a finite type surface S, introduced by Harvey in the late 1970s, 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 realized disjointly. The curve complex turned out to be a fundamental tool in the study of the geometry of the Teichmüller space
Teichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
, of mapping class group
Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
s and of Kleinian group
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s. In a 1999 paper 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
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
Ending lamination conjecture, a solution which was based on the combined work of Minsky, Masur, Brock and Canary. In 2006 Bowditch gave another proof 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 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
Mapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
Mod(S) on C(S) is "acylindrical" and that the asymptotic translation lengths of pseudo-anosov
Pseudo-Anosov map
In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus...
elements of Mod(S) on C(S) are rational numbers with bounded denominators.
A 2007 paper of Bowditch produces a positive solution of the angel problem
Angel problem
The angel problem is a question in game theory proposed by John Horton Conway. The game is commonly referred to as the Angels and Devils game. The game is played by two players called the angel and the devil. It is played on an infinite chessboard...
of John Conway
John Horton Conway
John Horton Conway is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory...
: Bowditch proved that a 4-angel has a winning strategy and can evade the devil in the "angel game". Independent solutions of the Angel problem
Angel problem
The angel problem is a question in game theory proposed by John Horton Conway. The game is commonly referred to as the Angels and Devils game. The game is played by two players called the angel and the devil. It is played on an infinite chessboard...
were produced at about the same time by Máthé and Kloster.
See also
- Geometric group theoryGeometric group theoryGeometric 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 .Another important...
- Geometric topologyGeometric topologyIn mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
- 3-manifold3-manifoldIn mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s - Kleinian groupKleinian groupIn mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
s
External links
- Brian H. Bowditch's HomePage at the University of WarwickUniversity of WarwickThe University of Warwick is a public research university located in Coventry, United Kingdom...