Large parameter spaces of quasiconformally distinct hyperbolic structures
The paper focuses on the isomorphisms between Fuchsian groups which are induced by self-mappings of the closed unit disc. Taking an infinite dimensional parameter space and embedding it into the Fenchel-Nielsen space of a surface \(S\), a real analytic embedding of this parameter space is obtained. This embedding is illustrated in an example of the bi-infinite flute surface having pants decomposition of type (3,0) with a puncture. The obtained image consists of quasiconformally non-conjugate, but topologically conjugate Fuchsian groups. These two conjugacies are compared to each other in particular cases. A Fuchsian group \(G\) is called flute if it is generated by \(G_i\), pairs of pants, with pairwise nonempty (in particular, equal to \(A(g_i+1)\) for \(g_{i+1}\) is a primitive hyperbolic boundary element both in \(G_i\) and \(G_{i+1})\) intersection of Nielsen convex regions \(N(G_i)\) and \(N(G_{i+1})\). It is shown that an isomorphism between two flute groups taking the the standard generators to standard generators is a topologic conjugacy. In particular the existence of an infinite dimensional set of the disjoint Teichmüller space close enough to a hyperbolic structure for a surface \(S\) is reduced. The paper continues the work on topological considerations of particular Fuchsian groups, see \textit{A. Marden} [Ann. Math., II. Ser. 99, 383-462 (1974; Zbl 0282.30014)]; [Adv. Compl. Funct. Theory, Proc. Semin. Maryland Univ. 1973/74, Lect. Notes Math. 505, 56-78 (1976; Zbl 0335.20024)); \textit{A. Basmajian} [Ann. Acad. Sci. Fenn., Ser. A I, 15, No. 1, 65-74 (1990; Zbl 0673.30032)]; [Trans. Am. Math. Soc. 336, No. 1, 421-444 (1993; Zbl 0768.30028)].
scientific article; zbMATH DE number 1038439
scientific article