The thick-thin decomposition and the bilipschitz classification of normal surface singularities

TL;DR

This paper introduces a natural decomposition of normal complex surface singularities into thick and thin parts, providing a complete bilipschitz classification based on topology and numerical invariants.

Contribution

It presents a novel thick-thin decomposition for surface singularities and characterizes their bilipschitz geometry using topological and numerical invariants.

Findings

The thin part is empty iff the singularity is metrically conical.

Most topology is concentrated in the thin parts.

The decomposition allows a complete bilipschitz classification.

Abstract

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of $(X,0)$ in terms of its topology and a finite list of numerical bilipschitz invariants.