Bieberbach-Auslander Theorem and Dynamics in Symmetric Spaces

The aim of this paper (my extended contribution to Intern. Conf. on Discrete Geometry dedicated to A. M. Zamorzaev) is to study dynamics of a discrete isometry group action in a noncompact symmetric space of rank one nearby its parabolic fixed points. Due to Margulis Lemma, such an action on corresponding horospheres is virtually nilpotent, so our extension of the Bieberbach-Auslander theorem for discrete groups acting on connected nilpotent Lie groups can be applied. As result, we show that parabolic fixed points of a discrete group of isometries of such symmetric space cannot be conical limit points and that the fundamental groups of geometrically finite locally symmetric of rank one orbifolds are finitely presented, and the orbifolds themselves are topologically finite.