Ending lamination theorem
Encyclopedia
In hyperbolic geometry
, the ending lamination theorem, originally conjectured by , states that hyperbolic 3-manifold
s with finitely generated fundamental group
s are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
The ending lamination theorem is a generalization of the Mostow rigidity theorem
to hyperbolic manifolds of infinite volume. When the manifold is compact or of finite volume, the Mostow rigidity theorem states that the fundamental group determines the manifold. When the volume is infinite the fundamental group is not enough to determine the manifold: one also needs to know the hyperbolic structure on the surfaces at the "ends" of the manifold, and also the ending laminations on these surfaces.
and proved the ending lamination conjecture for Kleinian
surface groups.
Suppose that a hyperbolic 3-manifold has a geometrically tame end of the form S×[0,1) for some compact surface S without boundary, so that S can be thought of as the "points at infinity" of the end. The ending lamination of this end is (roughly) a lamination on the surface S, in other words a closed subset of S that is written as the disjoint union of geodesics of S. It is characterized by the following property. Suppose that there is a sequence of closed geodesics on S whose lifts tends to infinity in the end. Then the limit of these simple geodesics is the ending lamination.
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, the ending lamination theorem, originally conjectured by , states that 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...
s 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...
s are determined by their topology together with certain "end invariants", which are geodesic laminations on some surfaces in the boundary of the manifold.
The ending lamination theorem is a generalization of the Mostow rigidity theorem
Mostow rigidity theorem
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique...
to hyperbolic manifolds of infinite volume. When the manifold is compact or of finite volume, the Mostow rigidity theorem states that the fundamental group determines the manifold. When the volume is infinite the fundamental group is not enough to determine the manifold: one also needs to know the hyperbolic structure on the surfaces at the "ends" of the manifold, and also the ending laminations on these surfaces.
and proved the ending lamination conjecture for Kleinian
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL 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...
surface groups.
Ending laminations
Ending laminations were introduced by .Suppose that a hyperbolic 3-manifold has a geometrically tame end of the form S×[0,1) for some compact surface S without boundary, so that S can be thought of as the "points at infinity" of the end. The ending lamination of this end is (roughly) a lamination on the surface S, in other words a closed subset of S that is written as the disjoint union of geodesics of S. It is characterized by the following property. Suppose that there is a sequence of closed geodesics on S whose lifts tends to infinity in the end. Then the limit of these simple geodesics is the ending lamination.