Kleinian groups with singly cusped parabolic fixed points

Abstract

We consider geometrically infinite Kleinian groups and, in particular, groups with singly cusped parabolic fixed points. In order to distinguish between different geometric characteristics of such groups, we introduce the notion of horospherical tameness. We give a brief discussion of the fractal nature of their limit sets. Subsequently, we use Jørgensen's analysis of punctured torus groups to give a canonical decomposition into ideal tetrahedra of the geometrically infinite end. This enables us to relate horospherical tameness to Diophantine properties of Thurston's end invariants.