Actions of discrete groups on spheres and real projective spaces (Q1017984)

A Cantor set \(K\subset S^n\) is tame if the homotopy groups \(\Pi_i(S^n-K)=0\), for \(0\leq i\leq n-2\) and the group \(\Pi_{n-1}(S^n-K)\) is infinitely generated. Otherwise \(K\) is wild. Note that if \(K\) is wild then there is no homeomorphism \(h:S^n\rightarrow S^n\) such that \(h(K)\) lies in a smoothly embedded arc. A classical example of a wild Cantor set is Antoine's necklace, see [\textit{D. Rolfsen}, Knots and links. 2nd print. with corr. Mathematics Lecture Series. 7. Houston, TX: Publish or Perish (1990; Zbl 0854.57002)]. In the paper under review it is shown that there exists a real analytic action of the free group \(F_8\) on \(S^3\), whose limit set \(\Lambda\) is a wild Cantor set. The action is proper, free, discontinuous and co-compact in \(S^3-\Lambda\). The set \(\Lambda\) is a kind of Antoine's necklace. This is generalized to odd dimensional spheres, and it is shown that there exist discrete, real analytic actions on \(S^{2n+1}\) whose limit set is a Cantor set wildly embedded in \(S^{2n+1}\). By an analogous construction, the authors define Schottky groups on real projective spaces of odd dimensions, and lift these actions to (locally) projective actions on \(S^{2n+1}\). By considering the quotient space of the domain of discontinuity by the group, new examples of manifolds with real projective structures are obtained.