The classification of kleinian groups of hausdorff dimension at most one

Abstract

Abstract In this paper, we provide the complete classification of 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 [4] is true: all non-classical Schottky groups must have Hausdorff dimension $\ge1$. The upper bounds of Hausdorff dimensions of classical Schottky groups has long been established by Phillips–Sarnak [14] and Doyle [8]. The proof of the theorem relies on the result of Bowen [3] and Hou [10].