hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...

, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by .

Earthquake maps

Given a simple closed

geodesic In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...

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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

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 William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ...

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.

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 was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

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