The classification of Kleinian groups of Hausdorff dimensions at most one

2016/10/10 by Hou, Yong
#53 #57 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1610.03046

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 \citeH then implies that, every closed Riemann surface is uniformizable by a classical Schottky group. The prove relie on the result of Hou \citeHou, and space of rectifiable \G-invariant closed curves.

Related

• The classification of Kleinian groups of Hausdorff dimensions at most one and Burnside's conjecture
• All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups
• Schottky uniformizations of Automorphisms of Riemann surfaces
• Towards a proof of the Classical Schottky Uniformization Conjecture
• The topology of Schottky spaces in higher dimensions