In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the tameness theorem states that every complete hyperbolic 3-manifold

Hyperbolic 3-manifold

A hyperbolic 3-manifold is a 3-manifold equipped with a complete Riemannian metric of constant sectional curvature -1. In other words, it is the quotient of three-dimensional hyperbolic space by a subgroup of hyperbolic isometries acting freely and properly discontinuously...

 with finitely generated fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 is topologically tame, in other words homeomorphic

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...

 to the interior of a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 3-manifold.

The tameness theorem was conjectured by . It was proved by and, independently, by Danny Calegari

Danny Calegari

Danny M. C. Calegari is the Richard Merkin Professor of Mathematics at the California Institute of Technology, and is one of the recipients of the 2009 Clay Research Award for his work in geometric topology and geometric group theory....

 and David Gabai

David Gabai

David Gabai, a mathematician, is currently a professor at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects....

. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the density theorem for Kleinian groups

Density theorem for Kleinian groups

In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups.-History:...

 and the ending lamination theorem

Ending lamination theorem

In hyperbolic geometry, the ending lamination theorem, originally conjectured by , 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...

.
It also implies the Ahlfors measure conjecture

Ahlfors measure conjecture

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....

.

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 surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

s. However, as the example of Alexander horned sphere

Alexander horned sphere

The Alexander horned sphere is a wild embedding of a sphere into space, discovered by . It is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:...

 shows, there are wild embeddings among 3-manifolds, so this property is not automatic.

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.

There had been steady progress in understanding tameness before the conjecture was resolved. Partial results had been obtained by Thurston

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...

, Brock, Bromberg, Canary, Evans, Minsky, Ohshika. An important sufficient condition for tameness in terms of splittings of the fundamental group had been obtained by Bonahon.

The conjecture was proved in 2004 by Ian Agol

Ian Agol

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

, 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".