Projective structures on Riemann surfaces and Kleinian groups
Let \(\Gamma\) be a Fuchsian universal covering group of a compact Riemann surface. Assume \(\Gamma\) acts on the upper half plane and let B(L,\(\Gamma)\) be the set of holomorphic functions \(\psi\) in the lower half plane which are holomorphic, which satisfy \(\psi (A(z))A'(z)^ 2=\psi (z)\) for all A in \(\Gamma\) and which have finite norm, \(\| \psi \| =\sup y^ 2| \psi (z)|\). If \(W_{\psi}\) is a holomorphic function in L whose Schwarzian derivative is equal to \(\psi\), then \(\psi\) determines a homomorphism \(\theta_{\psi}\) of \(\Gamma\) into PSL(2,\({\mathbb{C}})\) by the formula \[ w_{\psi}\circ A=\theta_{\psi}(A)\circ W_{\psi}\quad for\quad A\quad in\quad \Gamma. \] Let T(\(\Gamma)\) be the set of \(\psi\) for which \(W_{\psi}\) has a quasiconformal extension to the whole plane, let S(\(\Gamma)\) be the set of \(\psi\) such that \(W_{\psi}\) is schlicht in L and let K(\(\Gamma)\) be the set of \(\psi\) such that \(\Gamma^{\psi}=\theta_{\psi}(\Gamma)\) is a Kleinian group (with nonempty set of discontinuity in \({\mathbb{C}})\). The author proves the following theorems: Theorem 1. If \(\psi_ 0\) and \(\psi_ 1\) lie in a single component of Int(K(\(\Gamma)\)), then \(\Gamma^{\psi_ 1}\) and \(\Gamma^{\psi_ 2}\) are quasiconformally equivalent, that is, there exists a quasiconformal selfmapping w of \({\hat {\mathbb{C}}}\) such that \(w\circ A\circ w^{- 1}=\theta_{\psi_ 1}\circ \theta_{\psi_ 0}(A)\) for all A in \(\Gamma_{\psi_ 0}\). Moreover, if \(\| \psi_ 0-\psi_ 1\|\) is sufficiently small, then the quasiconformal selfmapping can be taken to have harmonic Beltrami coefficient. Theorem 2. For every \(\psi\) in Int(K(\(\Gamma)\)-T(\(\Gamma)\)), the function \(W_{\psi}\) is not a covering mapping on L. Consequently, \(W_{\psi}(L)={\hat {\mathbb{C}}}\). By definition, a b-group G is a group with only one simply connected invariant component \(\Delta\) in its set of discontinuity. Using the same formalism, one can construct a homomorphism \(\chi_{\psi}\) of G into PSL(2,\({\mathbb{C}})\) for every \(\psi\) in B(\(\Delta\),G). Theorem 3. Let G be a g-group with invariant component \(\Delta\). Then for each \(\epsilon >0\) there exists a \(\psi\) in \(B(\Delta,G)\) with norm less than \(\epsilon\) for which \(\chi_\psi(G)\) is not a Kleinian group.
scientific article; zbMATH DE number 4069280