Geometric infiniteness in negatively pinched Hadamard manifolds

TL;DR

This paper extends the understanding of geometrically infinite discrete subgroups from hyperbolic 3-space to more general negatively curved Hadamard manifolds, showing they have a rich set of limit points.

Contribution

It generalizes Bonahon's characterization to negatively pinched Hadamard manifolds, revealing properties of geometrically infinite subgroups in these spaces.

Findings

Geometrically infinite subgroups have continuum many nonconical limit points.

Extension of Bonahon's results to rank 1 symmetric spaces.

Results hold under bounded torsion assumption.

Abstract

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Every such geometrically infinite isometry subgroup $\Gamma$ has a set of nonconical limit points with cardinality of continuum.