Real and complex earthquakes

Abstract

We consider (real) earthquakes and, by their extensions, complex earthquakes of the hyperbolic plane H 2. We show that an earthquake restricted to the boundary S 1 of H 2 is a quasisymmetric map if and only if its earthquake measure is bounded. Multiplying an earthquake measure by a positive parameter we obtain an earthquake path. Consequently, an earthquake path with a bounded measure is a path in the universal Teichmüller space. We extend the real parameter for a bounded earthquake into the complex parameter with small imaginary part. Such obtained complex earthquake (or bending) is holomorphic in the parameter. Moreover, the restrictions to S 1 of a bending with complex parameter of small imaginary part is a holomorphic motion of S 1 in the complex plane. In particular, a real earthquake path with bounded earthquake measure is analytic in its parameter.