Higher dimensional complex Kleinian groups
Let \(N\) be an oriented Riemannian manifold of dimension \(2n\). The twistor space of \(N\) is the total space of a bundle over \(N\), whose fiber at each \(x\in N\) is the symmetric space \(SO(2n)/U(n)\), of all complex structures on \(T_x(N)\) which are compatible with the metric and the orientation of \(N\). The work of R. Penrose in the 1970's established a beautiful interplay between the conformal geometry of Riemannian manifolds and the complex geometry of their twistor spaces. In the article under review, the authors push forward these ideas and use twistor theory to construct new complex Kleinian groups in higher dimensions. One of their themes is that the conformal dynamics of Kleinian groups on spheres embeds in the holomorphic dynamics of complex Kleinian groups on projective spaces. They notice that if \(\Gamma\) is a discrete group of orientation-preserving conformal diffeomorphisms of \(N\), then \(\Gamma\) has a canonical lifting to a group \({\widetilde \Gamma}\) of diffeomorphisms of the twistor space of \(N\). They show that there is a natural limit set for the action of \({\widetilde \Gamma}\), which is the inverse image of the limit set of \(\Gamma\). They note an interesting feature here, which does not happen in the classical theory of limit sets of conformal Kleinian groups, which is the fact that the action of \({\widetilde \Gamma}\) on its limit set may or may not be minimal and ergodic with respect to the densities which come from the Patterson-Sullivan densities for \(\Gamma\). In the meantime, the authors prove a generalized version of a theorem which originates in the work of Sullivan on conformal dynamics and which says that in the case of the classical Kleinian groups, there is no invariant line field on the limit set. The authors' generalization says that every geometrically finite Zariski-dense subgroup of conformal maps of the sphere \(S^n\) acts minimally and ergodically on the \(p\)-dimensional subspaces of \(T(S^n)\) over the limit set, for every \(0<p<n\), and this is slightly more general than a previous result of Flaminio and Spatzier.
scientific article; zbMATH DE number 1747290