Dragomir Šarić

A measured laminations on the universal hyperbolic solenoid $\S$ is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid $\S$ is uniquely determined by a measured lamination on $\S$; it is a leafwise earthquake with the leafwise earthquake measure equal to the leafwise measured lamination. Leafwise earthquakes fit together to produce a new hyperbolic metric on $\S$ which is transversely continuous and we show that any two hyperbolic metrics on $\S$ are connected by an earthquake. We also establish the space of projective measured lamination $PML(\S)$ as a natural Thurston-type boundary to the Teichm\"uller space $T(\S)$ of the universal hyperbolic solenoid $\S$. The (baseleaf preserving) mapping class group $MCG_{BLP}(\S)$ acts continuously on the closure $T(\S)\cup PML(\S)$ of $T(\S)$. Moreover, the set of transversely locally constant measured laminations on $\S$ is dense in $ML(\S)$.

We give a presentation for the baseleaf preserving mapping class group $Mod(\S)$ of the punctured solenoid $\S$. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that $Mod(\S)$ has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which $Mod(\S)$ acts.

We consider (real) earthquakes and, by their extensions, complex earthquakes of the hyperbolic plane <!-- MATH: $\mathbb{H} ^2$ --> . We show that an earthquake restricted to the boundary of <!-- MATH: $\mathbb{H} ^2$ --> 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 of a bending with complex parameter of small imaginary part is a holomorphic motion of in the complex plane. In particular, a real earthquake path with bounded earthquake measure is analytic in its parameter.