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Coherence of limit points in the fibers over the asymptotic Teichmüller space. (English) Zbl 1322.30016

In the study of the Teichmüller space, the author introduced the limit set and the region of discontinuity for the Teichmüller modular group [Proc. Am. Math. Soc. 132, No. 1, 117–126 (2004; Zbl 1091.30012)]. In the paper under review the author considers a stationary subgroup of the quasiconformal mapping class group MCG(\(R\)) of a Riemann surface \(R\) and shows the coherence of the discreteness on each fiber of the projection from the Teichmüller space \(T(R)\) of \(R\) onto the asymptotic Teichmüller space \(AT(R)\) of \(R\). The asymptotic Teichmüller space of \(R\) is the set of all asymptotically equivalence classes of quasiconformal homeomorphisms of \(R\). It is represented by a quotient space of \(T(R)\). Let \(\pi: T(R) \to AT(R)\) be the projection and \(T_p\) the fiber of \(\pi(p)\) for a point \(p\) in \(T(R)\). For a subgroup \(G\) of MCG(\(R\)), \(G_*\) denotes the subgroup corresponding to \(G\) in the Teichmüller modular group Mod(\(R\)). The author shows that if \(G\) is stationary and closed, then for every point \(p\) in \(T(R)\), the fiber \(T_p\) is included in the limit set for \(G_*\) or in the region of discontinuity for \(G_*\). By giving an example, it is shown that the stationary property is necessary for the theorem. Furthermore the author makes some remarks on the theorem with respect to some previous results about Riemann surfaces satisfying the bounded geometry condition.

MSC:

30F60 Teichmüller theory for Riemann surfaces
30C62 Quasiconformal mappings in the complex plane
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)

Citations:

Zbl 1091.30012
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References:

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