In mathematics, the tameness theorem states that every complete

hyperbolic 3-manifold 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. I ...

with finitely generated

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

is topologically tame, in other words homeomorphic to the interior of a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

3-manifold. The tameness theorem was conjectured by . It was proved by and, independently, by

Danny Calegari Danny Matthew Cornelius Calegari is a mathematician who is currently a professor of mathematics at the University of Chicago. His research interests include geometry, dynamical systems, low-dimensional topology, and geometric group theory. Educ ...

and

David Gabai David Gabai is an American mathematician and 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. Biography David Gabai ...

. It is one of the fundamental properties of geometrically infinite hyperbolic 3-manifolds, together with the

density theorem for Kleinian groups In the mathematical theory of Kleinian groups, the density conjecture of Lipman Bers, Dennis Sullivan, and William Thurston, later proved independently by and , states that every finitely generated Kleinian group is an algebraic limit of geometri ...

and the

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 ge ...

. It also implies the

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. The conjecture was introduced by , who proved it in the case that the ...

History

Topological tameness may be viewed as a property of the

ends End, END, Ending, or variation, may refer to: End *In mathematics: **End (category theory) ** End (topology) ** End (graph theory) ** End (group theory) (a subcase of the previous) ** End (endomorphism) *In sports and games **End (gridiron footba ...

of the manifold, namely, having a local product structure. An analogous statement is well known in two dimensions, that is, for

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...

s. However, as the example of

Alexander horned sphere The Alexander horned sphere is a pathological object in topology discovered by . Construction The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...

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 Albert Marden (born 18 November 1934) is an American mathematician, specializing in complex analysis and hyperbolic geometry. Education and career Marden received his PhD in 1962 from Harvard University with thesis advisor Lars Ahlfors. Marden ha ...

, 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, 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 (born May 13, 1970) is an American mathematician who deals primarily with the topology of three-dimensional manifolds. Education and career Agol graduated with B.S. in mathematics from the California Institute of Technology in 1992 and ...

, 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".

References

* . * . * * . * * . {{Manifolds 3-manifolds Conjectures that have been proved Differential geometry Hyperbolic geometry Kleinian groups Manifolds Theorems in geometry

History

See also

References