hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P' ...

, an earthquake map is a method of changing one

hyperbolic manifold In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, re ...

into another, introduced by .

Earthquake maps

Given a simple closed

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

on an oriented hyperbolic surface 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 surface, 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 it. 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 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 ''E'E'' is a hyperbolic transformation whose axis separates ''A'' and ''B'' and which translates to the left, where ''E''_''A'' 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 a

Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...

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 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 Given an oriented surface, we can divide the group Diff(''S''), the gr ...

References

* * *{{citation , chapter=Earthquakes in two-dimensional hyperbolic geometry , title=Low dimensional topology and Kleinian groups , first=William P. , last=Thurston , authorlink=William Thurston, editor=D.B.A. Epstein , year=1986 , publisher=

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

, isbn=978-0-521-33905-6 Hyperbolic geometry Functions and mappings