Hot! Earthquake Map

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In earthquake map is a method of changing one hyperbolic manifold into another, introduced by 1986 ).

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.

Ageodesic laminationis of a hyperbolic surface is a closed subset with a foliation by geodesics. Aleft earthquakeE 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 1 AE B 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 Kerckhoff (1983) , who used it to solve the Nielsen realization problem .

References

Kerckhoff, Steven P. (1983), "The Nielsen realization problem", Annals of Mathematics. Second Series117(2): 235 265, doi :10.2307/2007076 , ISSN 0003-486X , 690845

66, Paris:

Low dimensional topology and Kleinian groups,