Earthquake map
Encyclopedia
In hyperbolic geometry
, an earthquake map is a method of changing one hyperbolic manifold
into another, introduced by .
More generally one can do the same construction with a finite number of disjoint simple geodesics, each with a real number attached to them. The result is called a simple earthquake.
An earthquake is roughly a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic one puts a measure on them.
A geodesic lamination is of a hyperbolic surface is a closed subset with a foliation by geodesics. A left earthquake E consists of a map between copies of the hyperbolic plane with geodesic laminations, that is an isometry from each stratum of the foliation to a stratum. Moreover if A and B are two strata then EE is a hyperbolic transformation whose axis separates A and B and which translates to the left, where EA is the isometry of the whole plane that restricts to E on A, and likewise for B.
there is a unique left earthquake from x to y. It was proved by William Thurston in a course in Princeton in 1976-1977, but at the time he did not publish it, and the first published statement and proof was given by , who used it to solve the Nielsen realization problem
.
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
, an earthquake map is a method of changing one hyperbolic manifold
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
into another, introduced by .
Earthquake maps
Given a simple geodesic on an oriented hyperbolic manifold and a real number t, one can cut the manifold along the geodesic, slide the edges a distance t to the left, and glue them back. This gives a new hyperbolic manifold, and the (possibly discontinuous) map between them is an example of a left earthquake.More generally one can do the same construction with a finite number of disjoint simple geodesics, each with a real number attached to them. The result is called a simple earthquake.
An earthquake is roughly a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic one puts a measure on them.
A geodesic lamination is of a hyperbolic surface is a closed subset with a foliation by geodesics. A left earthquake E consists of a map between copies of the hyperbolic plane with geodesic laminations, that is an isometry from each stratum of the foliation to a stratum. Moreover if A and B are two strata then EE is a hyperbolic transformation whose axis separates A and B and which translates to the left, where EA is the isometry of the whole plane that restricts to E on A, and likewise for B.
Earthquake theorem
Thurston's earthquake theorem states that for any two points x, y of Teichmüller spaceTeichmüller space
In mathematics, the Teichmüller space TX of a topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism...
there is a unique left earthquake from x to y. It was proved by William Thurston in a course in Princeton in 1976-1977, but at the time he did not publish it, and the first published statement and proof was given by , who used it to solve the Nielsen realization problem
Nielsen realization problem
The Nielsen realization problem is a question asked by about whether finite subgroups of mapping class groups can act on surfaces, that was answered positively by .-Statement:...
.