Real and complex earthquakes. (English) Zbl 1094.30049
W. P. Thurston introduced the (left, real) earthquake as a method for moving from one hyperbolic structure on a surface to any other: one splits the surface along the leaves (fault lines) of an appropriate geodesic lamination (the fault zone), shifts the surface to the left along the fault lines a distance dictated by a transverse measure on the lamination, and then reglues. This discontinuous transformation, when lifted to the universal cover \(H^2\), extends continuously to a homeomorphism (determined up to Möbius transformation) of the circle at infinity. (Thurston: Geology is transitive, in the sense that all orientation preserving homeomorphisms of the circle are realizable.) This homeomorphism at infinity determines a new hyperbolic structure on the surface (unique up to equivariant isotopy that is the identity at infinity).
The earthquake has proved a useful tool in the study of Teichmüller space, as, for example, in Kerckhoff’s proof of the Nielsen realization conjecture. Complex earthquakes have proved useful in the study of Kleinian groups, with the imaginary parameter serving to express the bending of a copy of \(H^2\) in \(H^3\) along the fault lines.
The author extends a number of results, originally proved for the Teichmüller space of closed surfaces, to universal Teichmüller space. The corresponding results for finite surfaces can again be deduced from this more general case, but the new results apply also to noncompact surfaces.
Among the main results are these: (1) The map on the circle \(S^1\) at infinity is quasisymmetric, hence corresponds to a quasiconformal change of hyperbolic structure, if and only if the transverse measure on the lamination is uniformly bounded on transversals of length \(\leq 1\). (2) Small bending results in a convex surface. (3) Kerckhoff’s result on the analyticity of earthquake paths for finite surfaces and McMullen’s result on the holomorphicity of complex earthquakes extend to arbitrary hyperbolic surfaces.
The earthquake has proved a useful tool in the study of Teichmüller space, as, for example, in Kerckhoff’s proof of the Nielsen realization conjecture. Complex earthquakes have proved useful in the study of Kleinian groups, with the imaginary parameter serving to express the bending of a copy of \(H^2\) in \(H^3\) along the fault lines.
The author extends a number of results, originally proved for the Teichmüller space of closed surfaces, to universal Teichmüller space. The corresponding results for finite surfaces can again be deduced from this more general case, but the new results apply also to noncompact surfaces.
Among the main results are these: (1) The map on the circle \(S^1\) at infinity is quasisymmetric, hence corresponds to a quasiconformal change of hyperbolic structure, if and only if the transverse measure on the lamination is uniformly bounded on transversals of length \(\leq 1\). (2) Small bending results in a convex surface. (3) Kerckhoff’s result on the analyticity of earthquake paths for finite surfaces and McMullen’s result on the holomorphicity of complex earthquakes extend to arbitrary hyperbolic surfaces.
Reviewer: James W. Cannon (Provo)
MSC:
| 30F60 | Teichmüller theory for Riemann surfaces |
| 30C62 | Quasiconformal mappings in the complex plane |
| 30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |
References:
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