The classification of Kleinian groups of Hausdorff dimensions at most one and Burnside's conjecture

TL;DR

This paper classifies convex cocompact Kleinian groups with Hausdorff dimension less than one, showing they are all classical Schottky groups, and confirms the converse of Burnside's conjecture.

Contribution

It provides a complete classification of such Kleinian groups and proves the sharpness of the Hausdorff dimension bound, confirming the converse of Burnside's conjecture.

Findings

Convex cocompact Kleinian groups with Hausdorff dimension <1 are classical Schottky groups.

The Hausdorff dimension bound of 1 is sharp.

The converse of Burnside's conjecture is validated.

Abstract

In this paper we provide the complete classification of convex cocompact Kleinian group of Hausdorff dimensions less than $1.$ In particular, we prove that every convex cocompact Kleinian group of Hausdorff dimension $<1$ is a classical Schottky group. This upper bound is sharp. The result implies that the converse of Burside's conjecture \cite{Burside} is true: All non-classical Schottky groups must have Hausdorff dimension $\ge1$. The prove of the theorem relies on the result of Hou \cite{Hou}.