Acta Mathematica

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

Lev Birbrair, Walter D. Neumann, and Anne Pichon

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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.

Article information

Acta Math., Volume 212, Number 2 (2014), 199-256.

Received: 19 January 2012
Revised: 21 June 2013
First available in Project Euclid: 30 January 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
Secondary: 32S25: Surface and hypersurface singularities [See also 14J17] 32S05: Local singularities [See also 14J17] 57M99: None of the above, but in this section

bilipschitz geometry normal surface singularity thick-thin decomposition

2014 © Institut Mittag-Leffler


Birbrair, Lev; Neumann, Walter D.; Pichon, Anne. The thick-thin decomposition and the bilipschitz classification of normal surface singularities. Acta Math. 212 (2014), no. 2, 199--256. doi:10.1007/s11511-014-0111-8.

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