Discrete actions on nilpotent Lie groups and negatively curved spaces

Abstract

The aim of this paper is to study dynamics of a discrete isometry group action in a pinched Hadamard manifold nearby its parabolic fixed points. Due to Margulis Lemma, such an action on corresponding horospheres is virtually nilpotent, so we solve the problem by establishing a structural theorem for discrete groups acting on connected nilpotent Lie groups. As applications, we show that parabolic fixed points of a discrete isometry group cannot be conical limit points, that the fundamental groups of geometrically finite orbifolds with pinched negative sectional curvature are finitely presented, and the orbifolds themselves are topologically finite.