Hausdorff dimension of non-conical limit sets

TL;DR

This paper investigates the Hausdorff dimension of non-conical limit sets in Kleinian groups, showing it can be zero for certain groups but positive for finitely generated ones, advancing understanding of their geometric properties.

Contribution

It constructs a specific geometrically infinite Fuchsian group with zero Hausdorff dimension for its nonconical limit set and proves positivity for finitely generated cases.

Findings

Existence of a geometrically infinite Fuchsian group with zero Hausdorff dimension

Positivity of Hausdorff dimension for finitely generated, geometrically infinite Kleinian groups

Insights into the geometric structure of non-conical limit sets

Abstract

Geometrically infinite Kleinain groups have nonconical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the nonconical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the nonconical limit set is positive.