On horospheric limit sets of Kleinian groups

TL;DR

This paper investigates the size of the difference between big and small horospheric limit sets of Kleinian groups, showing it has zero conformal measure in certain cases, and explores the relation with Myrberg limit sets.

Contribution

It provides new insights into the measure-theoretic properties of horospheric limit sets for normal subgroups of Kleinian groups of divergence type.

Findings

The difference between big and small horospheric limit sets has zero conformal measure.

The Myrberg limit set is contained in the horospheric limit set of any non-trivial normal subgroup.

The results partially answer Tukia's question on the size of these limit set differences.

Abstract

In this paper we partially answer a question of P. Tukia about the size of the difference between the big horospheric limit set and the horospheric limit set of a Kleinian group. We mainly investigate the case of normal subgroups of Kleinian groups of divergence type and show that this difference is of zero conformal measure by using another result obtained here: the Myrberg limit set of a non-elementary Kleinian group is contained in the horospheric limit set of any non-trivial normal subgroup.