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

Abstract

International audienceWe 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

Thin-thick decomposition for real definable isolated singularities

“…Finally we show that the topology of the thin-thick decomposition of the set, equipped with the multiplicity, is an invariant for the blow-spherical equivalence. We observe that the thin-thick decomposition from [7] is a particular case of our construction.…”

“…The Thin-Thick Decomposition from [7] is obtained from our ThinThick Decomposition in the following way. Consider a normal complex surface singularity and consider its real normal embedding.…”

“…Recent investigations in Lipschitz Geometry of complex algebraic surface singularities are closely related to the notion of thin-thick decomposition discovered in [7]. Indeed a germ of a normal complex algebraic surface can be divided into two zones.…”

“…The other zone, called the thin zone, is not metrically conical, has density zero at the origin and moreover contains some of the most important invariants of the Lipschitz geometry of singular subset germs such as fast loops, choking-horns or separating sets ( [4,5,3]). The topology of the thin-thick decomposition is Lipschitz invariant and moreover the complete inner Lipschitz invariant of normal complex algebraic surface singularity germs can be obtained as the iterated thin-thick decomposition [7].…”

“…We introduce the thin-thick decomposition of an isolated singularity germ -which happens to be a natural blow-spherical invariant. This decomposition is a generalization of the thin-thick decomposition of normal complex surface singularity germs introduced in [7]. …”