Tameness theorem

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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.

Mathematics field of study

Mathematics includes the study of such topics as quantity, structure, space, and change.

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.

In the mathematical field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.

The tameness theorem was conjectured by Marden (1974). It was proved by Agol (2004) and, independently, by Danny Calegari and David Gabai. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups and the ending lamination theorem. It also implies the Ahlfors measure conjecture.

Danny M. C. Calegari is an Australian-American mathematician who is currently a Professor at the University of Chicago. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory. Calegari was one of the recipients of the 2009 Clay Research Award for his solution to the Marden Tameness Conjecture and the Ahlfors Measure Conjecture.

David Gabai, a mathematician, is the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects.

In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved by Namazi & Souto (2010) and Ohshika (2011), states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.

History

Topological tameness may be viewed as a property of the ends of the manifold, namely, having a local product structure. An analogous statement is well known in two dimensions, i.e. for surfaces. However, as the example of Alexander horned sphere shows, there are wild embeddings among 3-manifolds, so this property is not automatic.

In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification.

Surface (topology) two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space

In topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

Alexander horned sphere

The Alexander horned sphere is a pathological object in topology discovered by J. W. Alexander (1924).

The conjecture was raised in the form of a question by Albert Marden, who proved that any geometrically finite hyperbolic 3-manifold is topologically tame. The conjecture was also called the Marden conjecture or the tame ends conjecture.

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

There had been steady progress in understanding tameness before the conjecture was resolved. Partial results had been obtained by Thurston, Brock, Bromberg, Canary, Evans, Minsky, Ohshika.[ citation needed ] An important sufficient condition for tameness in terms of splittings of the fundamental group had been obtained by Bonahon.[ citation needed ]

William Thurston mathematician

William Paul Thurston was an American mathematician. He was 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. From 2003 until his death he was a professor of mathematics and computer science at Cornell University.

Francis Bonahon French mathematician

Francis Bonahon is a French mathematician, specializing in low-dimensional topology.

The conjecture was proved in 2004 by Ian Agol, and independently, by Danny Calegari and David Gabai. Agol's proof relies on the use of manifolds of pinched negative curvature and on Canary's trick of "diskbusting" that allows to replace a compressible end with an incompressible end, for which the conjecture has already been proved. The Calegari–Gabai proof is centered on the existence of certain closed, non-positively curved surfaces that they call "shrinkwrapped".

Ian Agol American mathematician

Ian Agol is an American mathematician who deals primarily with the topology of three-dimensional manifolds.

Related Research Articles

Low-dimensional topology branch of topology that studies topological spaces of four or fewer dimensions

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. It 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.

3-manifold 3-dimensional manifold

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional 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 enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

Kleinian group discrete group of Möbius transformations

In mathematics, a Kleinian group is a discrete subgroup of PSL(2, C). The group PSL(2, C) 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 space H3, and as orientation preserving conformal maps of the open unit ball B3 in R3 to itself. Therefore, a Kleinian group can be regarded as a discrete subgroup acting on one of these spaces.

In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifolds by Mostow (1968) and extended to finite volume manifolds by Marden (1974) in 3 dimensions, and by Prasad (1973) in all dimensions at least 3. Gromov (1981) gave an alternate proof using the Gromov norm.

In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.

In mathematics, hyperbolic Dehn surgery is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.

In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover that is a Haken manifold.

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.

In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space by an arithmetic Kleinian group. These manifolds include some particularly beautiful or remarkable examples.

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.

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, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0.

Jeffrey Brock 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.

References

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