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

Lev Birbrair Departamento de Matemática, Universidade Federal do Ceará Walter D. Neumann Department of Mathematics, Barnard College, Columbia University Anne Pichon Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, Marseille, France

Complex Variables and Complex Analysis mathscidoc:1701.01006

Acta Mathematica, 212, (2), 199-256, 2012.1
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.
bilipschitz geometry; normal surface singularity; thick-thin decomposition
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@inproceedings{lev2012the,
  title={The thick-thin decomposition and the bilipschitz classification of normal surface singularities},
  author={Lev Birbrair, Walter D. Neumann, and Anne Pichon},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402681813787},
  booktitle={Acta Mathematica},
  volume={212},
  number={2},
  pages={199-256},
  year={2012},
}
Lev Birbrair, Walter D. Neumann, and Anne Pichon. The thick-thin decomposition and the bilipschitz classification of normal surface singularities. 2012. Vol. 212. In Acta Mathematica. pp.199-256. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402681813787.
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