mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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. It ...

with finitely generated

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

is topologically tame, in other words

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

3-manifold. The tameness theorem was

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), ha ...

d by . It was proved by and, independently, by

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

and

David Gabai David Gabai is an American mathematician and the Princeton University Department of Mathematics, Hughes-Rogers Professor of Mathematics at Princeton University. His research focuses on low-dimensional topology and hyperbolic geometry. Biography ...

. 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 In mathematics, a Kleinian gr ...

and the

ending lamination theorem In hyperbolic geometry, the ending lamination theorem, originally conjectured by as the eleventh problem out of his twenty-four questions, states that hyperbolic 3-manifolds with finitely generated fundamental groups are determined by their topol ...

. It also implies the Ahlfors measure conjecture.

History

Topological tameness may be viewed as a property of the

ends End, END, Ending, or ENDS may refer to: End Mathematics *End (category theory) *End (topology) *End (graph theory) *End (graph_theory)#Cayley_graphs, End (group theory) (a subcase of the previous) *End (endomorphism) Sports and games *End (gridir ...

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 ...

s. However, as the example of

Alexander horned sphere The Alexander horned sphere is a pathological object in topology discovered by . It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topological 3-ball, the Alexande ...

shows, there are wild embeddings among , 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 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 Thurston may refer to: Places Antarctica * Thurston Glacier, Marie Byrd Land * Thurston Island, off Ellsworth Land United Kingdom * Thurston, Suffolk, England, a village and parish ** Thurston railway station United States * Thurston County, Neb ...

, 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 a ...

, 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 Hyperbolic geometry Kleinian groups Theorems in differential geometry

History

See also

References