Dragomir Šarić

Real and complex earthquakes

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 Teichmiiller 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.

Teichmüller mapping class group of the universal hyperbolic solenoid

We show that the homotopy class of a quasiconformal self-map of the universal hyperbolic solenoid H_∞ is the same as its isotopy class and that the uniform convergence of quasiconformal self-maps of H_∞ to the identity forces them to be homotopic to conformal maps. We identify a dense subset of T(H_∞) such that the orbit under the base leaf preserving mapping class group MCG_(BLP)(H_∞) of any point in this subset has accumulation points in the Teichmuller space T(H_∞). Moreover, we show that finite subgroups of MCG_(BLP)(H_∞) are necessarily cyclic and that each point of T(H_∞) has an infinite isotropy subgroup in MCG_(BLP)(H_∞).

Infinitesimal Liouville Distributions for Teichmüller Space

We consider an arbitrary Riemann surface

Complex Fenchel-Nielsen coordinates with small imaginary parts

X

A presentation for the baseleaf preserving mapping class group of the punctured solenoid

, possibly of infinite hyperbolic area. The Liouville measure of the hyperbolic metric defines a measure on the space

Circle homeomorphisms and shears

G(\tilde{X})

Extremal metrics and modulus

of geodesics of the universal covering

Bendings by finitely additive transverse cocycles

\tilde{X}

Barycentric extension and the Bers embedding for asymptotic Teichmüller space

of

Geodesic currents and Teichmüller space

X