On the absence of the Ahlfors and Sullivan theorems for Kleinian groups in higher dimensions (Q1177547)

The authors present a slight modification of their construction [Topology Appl. 40, No. 1, 83--91 (1991; Zbl 0817.57018)] of ``wild'' Kleinian groups in 3-sphere. The constructed finitely generated groups \(G\) with invariant discontinuity set components \(\Omega_ 0\) have the manifolds \(\Omega_ 0/G\) with infinitely generated fundamental group and infinitely many cusps. This shows the effect in the title, for the case of non-contractible discontinuity set components. The construction is based on two main ideas. First idea is the ``block-building'' construction (using a finite number of Maskit combinations) of finitely generated ``wild'' Kleinian groups \(G\) with infinitely generated fundamental group of invariant discontinuity set components, \(G=\ldots * G_i*_{G_{ij}}G_j*\). That method was developed long ago [\textit{B. N. Apanasov} and \textit{A. V. Tetenov}, J. Diff. Geom. 28, 407--422 (1988; Zbl 0627.57015); B. N. Apanasov, Ann. Acad. Sci. Fenn., Ser. A I, Math. 14, No. 2, 243--255 (1989; Zbl 0747.30016)] (the authors use it without any reference). Second idea of the authors construction is the using, as the mentioned groups \(G_i\), some special finitely generated groups isomorphic to surface bundle groups. As in the above references, that groups \(G_ i\) preserve the block-balls \(B_i\subset S^3\) organizing some finite beads. See also \textit{L. V. Ahlfors} [Am. J. Math. 86, 413--429 (1964; Zbl 0133.04201)] and \textit{D. Sullivan} [ Acta Math. 147, 289-299 (1981; Zbl 0502.57004)].