Earthquake map
{{Short description|Concept in hyperbolic geometry}}
{{For|maps of literal earthquakes|Seismic hazard map}}
In hyperbolic geometry, an earthquake map is a method of changing one hyperbolic manifold into another, introduced by {{harvs|txt|authorlink=William Thurston|first=William|last=Thurston|year=1986}}.
Earthquake maps
Given a simple closed geodesic 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{{su|b=A|p=−1}}E{{su|b=B|p=}} 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 a Teichmüller space 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 {{harvtxt|Kerckhoff|1983}}, who used it to solve the Nielsen realization problem.
References
- {{Citation | last1=Kerckhoff | first1=Steven P. | title=The Nielsen realization problem |authorlink=Steven Kerckhoff| doi=10.2307/2007076 | mr=690845 | year=1983 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=117 | issue=2 | pages=235–265| jstor=2007076 | citeseerx=10.1.1.353.3593 }}
- {{Citation | title=Travaux de Thurston sur les surfaces | url=https://books.google.com/books?id=uxEZAQAAIAAJ | publisher=Société Mathématique de France | location=Paris | series=Astérisque | isbn=978-99920-1-230-7 | mr=568308 | year=1979 | volume=66}}
- {{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 | isbn=978-0-521-33905-6 }}