arXiv subjects
Publications and source records attributed to Dragomir Saric.
Let $M$ be a closed surface. By $\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says that for any closed surface $M$ of genus $\g \ge 2$, there is no homomorphic section $\E:\MC(M) \to \Homeo(M)$ of the standard projection map $\Proj:\Homeo(M) \to \MC(M)$.
For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+ε)$-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichmüller space $T(§)$ of the punctured solenoid $§$ under the action of the corresponding Modular group (which is the mapping class group of $§$ \cite{NS}, \cite{Odd}) has the closure in $T(§)$ strictly larger than the orbit and that the closure is necessarily uncountable.
We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.
A measured laminations on the universal hyperbolic solenoid $§$ is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid $§$ is uniquely determined by a measured lamination on $§$; 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 $§$ which is transversely continuous and we show that any two hyperbolic metrics on $§$ are connected by an earthquake. We also establish the space of projective measured lamination $PML(§)$ as a natural Thurston-type boundary to the Teichmüller space $T(§)$ of the universal hyperbolic solenoid $§$. The (baseleaf preserving) mapping class group $MCG_{BLP}(§)$ acts continuously on the closure $T(§)\cup PML(§)$ of $T(§)$. Moreover, the set of transversely locally constant measured laminations on $§$ is dense in $ML(§)$.
The punctured solenoid $§$ is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of $§$ is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of $§$. Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of $§$ giving a combinatorial description of its decorated Teichmüller space itself. This is used to obtain a non-trivial set of generators of the modular group of $§$, which is presumably the main result of this paper. Moreover, each word in these generators admits a normal form, and the natural equivalence relation on normal forms is described. There is furthermore a non-degenerate modular group invariant two form on the Teichmüller space of $§$. All of this structure is in perfect analogy with that of the decorated Teichmüller space of a punctured surface of finite type.