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

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

2011-05-17

arxiv.org/abs/1105.3327v3

Mathematics

Algebraic Geometry

v3 is a minor revision over v2: Two words were omitted in the previous version from the sentence in Theorem 1.9: "The bilipsch

Scientific paper

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.

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