The classification of Kleinian groups of Hausdorff dimensions at most one

TL;DR

This paper completely classifies Kleinian groups with Hausdorff dimension less than 1, showing they are classical Schottky groups, and applies this to uniformization of Riemann surfaces.

Contribution

It provides the first complete classification of Kleinian groups with Hausdorff dimension under 1, establishing a sharp upper bound and linking to classical Schottky groups.

Findings

Purely loxodromic Kleinian groups with dim<1 are classical Schottky groups

The upper bound of 1 is sharp for Hausdorff dimension

Every closed Riemann surface can be uniformized by a classical Schottky group

Abstract

In this paper we provide the complete classification of Kleinian groups of Hausdorff dimensions less than $1.$ In particular, we prove that every purely loxodromic Kleinian groups of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. As an application, the result of \cite{H} then implies that, every closed Riemann surface is uniformizable by a classical Schottky group. The prove relie on the result of Hou \cite{Hou}, and space of rectifiable $\G$-invariant closed curves.