mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the theorem of

Gromov Gromov (russian: Громов) is a Russian male surname, its feminine counterpart is Gromova (Громова). Gromov may refer to: * Alexander Georgiyevich Gromov (born 1947), Russian politician and KGB officer * Alexander Gromov (born 1959), R ...

and Thurston states a sufficient condition for

Dehn filling 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 hyperb ...

on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let be a cusped hyperbolic 3-manifold. Disjoint

horoball In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space. It is the boundary of a horoball, the limit of a sequence of increasing balls sharing (on one side) a tangent hyperplane and its point of ...

neighborhoods of each cusp can be selected. The boundaries of these neighborhoods are quotients of horospheres and thus have Euclidean metrics. A slope, i.e. unoriented isotopy class of simple closed curves on these boundaries, thus has a well-defined length by taking the minimal Euclidean length over all curves in the isotopy class. The theorem states: a Dehn filling of with each filling slope greater than results in a 3-manifold with a complete metric of negative sectional curvature. In fact, this metric can be selected to be identical to the original hyperbolic metric outside the horoball neighborhoods. The basic idea of the proof is to explicitly construct a negatively curved metric inside each horoball neighborhood that matches the metric near the horospherical boundary. This construction, using cylindrical coordinates, works when the filling slope is greater than . See for complete details. According to the geometrization conjecture, these negatively curved 3-manifolds must actually admit a complete hyperbolic metric. A horoball packing argument due to Thurston shows that there are at most 48 slopes to avoid on each cusp to get a hyperbolic 3-manifold. For one-cusped hyperbolic 3-manifolds, an improvement due to Colin Adams gives 24 exceptional slopes. This result was later improved independently by and with the 6 theorem. The "6 theorem" states that Dehn filling along slopes of length greater than 6 results in a ''hyperbolike'' 3-manifold, i.e. an

irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

atoroidal In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus. There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non- boundary parallel, incompressible t ...

, non-

Seifert-fibered A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for a ...

3-manifold with infinite word hyperbolic

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 ...

. Yet again assuming the geometrization conjecture, these manifolds have a complete hyperbolic metric. An argument of Agol's shows that there are at most 12 exceptional slopes.

References

*. *. *. {{DEFAULTSORT:2pi Theorem 3-manifolds Theorems in geometry