Actions of discrete groups on spheres and real projective spaces

Abstract

In this paper, we first define discrete, smooth actions on $$ \mathbb{S}^{{2n + 1}} $$ , whose limit sets are Cantor sets wildly embedded in $$ \mathbb{S}^{{2n + 1}} $$ (Antoine's necklaces). Secondly, we define Schottky groups on real projective spaces of odd dimensions, $$ \mathbb{P}^{{2n + 1}}_{\mathbb{R}} $$ . We lift these actions to (locally) projective actions on the sphere $$ \mathbb{S}^{{2n + 1}} $$ and consider the quotient space of the domain of discontinuity by the group to obtain new examples of manifolds with real projective structures.