The geometry of Nielsen's hull for a Kleinian group in space and quasi- conformal mappings
The author gives a method of investigation of the geometry of the convex hull \(H_ G\) (in the hyperbolic space \(H^{n+1})\) of the limit set \(L(G)\subset \partial H^{n+1}=S^ n\) for a discrete Möbius group \(G\subset Moeb(n)\), \(n\geq 2\). The essence of the method is that he considers the covering of the discontinuity set \(O(G)\subset S^ n\) by so-called maximal balls. The case \(n\geq 3\) differs from \(n=2\) which was investigated by W. Thurston and detailed by \textit{D. B. A. Epstein} and \textit{A. Marden} [Analytical and geometric aspects of hyperbolic space, Symp. Warwick and Durham/Engl. 1984, Lond. Math. Soc. Lect. Note Ser. 111, 113-254 (1987; Zbl 0612.57010)]. It turns out, in particular, that for \(n\geq 3\) the boundary \(\partial H_ G\) cannot generally admit an isometric development in \(H^ n\). That disproves the Kourouniotis- Johnson-Millson conjecture: there exist deformations of the conformal structure of \(H^ n/G\), vol\(H^ n/G<\infty\), \(n\geq 3\), distinct from bending deformations [see the author, Ann. Math. Stud. 97, 21-31 (1981; Zbl 0464.30037); Dokl. Akad. Nauk SSSR 293, 14-17 (1987; Zbl 0633.57008); and \textit{Chr. Kourouniotis}, Math. Proc. Camb. Philos. Soc. 98, 247-261 (1985; Zbl 0577.53041)]. As the main application the author constructs a K-quasi-conformal G- equivariant map \(F: O(G)\to \partial H_ G\) for a geometrically finite (for \(n=2\) finitely generated) group \(G\subset Moeb(n)\) whose limit set is not contained in an (n-1)-sphere. For a quasi-Fuchsian group \(G\subset Moeb(2)\) this is a theorem of D. Sullivan, proved by Epstein and Marden [loc. cit.].
Kourouniotis-\(Johnson\)- \(Millson\) conjecture
scientific article; zbMATH DE number 4082443
scientific article