Rigidity of limit sets for nonplanar geometrically finite Kleinian groups of the second kind

TL;DR

This paper explores the rigidity properties of limit sets for nonplanar geometrically finite Kleinian groups in infinite-dimensional hyperbolic space, extending some classical theorems but not others.

Contribution

It generalizes Susskind and Swarup's rigidity theorem to infinite dimensions, while demonstrating the limitations of Yang and Jiang's stronger rigidity theorem.

Findings

Susskind and Swarup's rigidity theorem extends to infinite dimensions.

Yang and Jiang's stronger rigidity theorem does not hold in infinite dimensions.

The paper clarifies the boundaries of rigidity phenomena in infinite-dimensional hyperbolic geometry.

Abstract

We consider the relation between geometrically finite groups and their limit sets in infinite-dimensional hyperbolic space. Specifically, we show that a rigidity theorem of Susskind and Swarup ('92) generalizes to infinite dimensions, while a stronger rigidity theorem of Yang and Jiang ('10) does not.