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 Mosc. Math. J., 2003, Volume 3, Number 2, Pages 273–333 (Mi mmj89)

The combinatorial geometry of singularities and Arnold's series $E$$Z$$Q$

E. Brieskorn, A. M. Pratusevich, F. Rothenhäusler

University of Bonn, Institute for Applied Mathematics

Abstract: We consider discrete subgroups $\Gamma$ of the simply connected Lie group $\widetildeSU(1,1)$ of finite level. This Lie group has the structure of a 3-dimensional Lorentz manifold coming from the Killing form. $\Gamma$ acts on $\widetildeSU(1,1)$ by left translations. We want to describe the Lorentz space form $\Gamma\setminus\widetildeSU(1,1)$ by constructing a fundamental domain $F$ for $\Gamma$. We want $F$ to be a polyhedron with totally geodesic faces. We construct such $F$ for all $\Gamma$ satisfying the following condition: The image $\overline\Gamma$ of $\Gamma$ in $PSU(1,1)$ has a fixed point $u$ in the unit disk of order larger than the level of $\Gamma$. The construction depends on $\Gamma$ and $\Gamma u$.
For co-compact ${\Gamma}$ the Lorentz space form $\Gamma\setminus\widetildeSU(1,1)$ is the link of a quasi-homogeneous Gorenstein singularity. The quasi-homogeneous singularities of Arnold's series $E$$Z$$Q$ are of this type. We compute the fundamental domains for the corresponding group. They are represented by polyhedra in Lorentz 3-space shown on Tables 1–13. Each series exhibits a regular characteristic pattern of its combinatorial geometry related to classical uniform polyhedra.

Key words and phrases: Lorentz space form, polyhedral fundamental domain, quasihomogeneous singularity, Arnold singularity series.

Full text: http://www.ams.org/.../abst3-2-2003.html
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Bibliographic databases:

MSC: Primary 53C50; Secondary 14J17, 20H10, 30F35, 30F60,32G15, 32S25, 51M20, 52
Language: English

Citation: E. Brieskorn, A. M. Pratusevich, F. Rothenhäusler, “The combinatorial geometry of singularities and Arnold's series $E$$Z$$Q$”, Mosc. Math. J., 3:2 (2003), 273–333

Citation in format AMSBIB
\Bibitem{BriPraRot03} \by E.~Brieskorn, A.~M.~Pratusevich, F.~Rothenh\"ausler \paper The combinatorial geometry of singularities and Arnold's series~$E$,~$Z$,~$Q$ \jour Mosc. Math.~J. \yr 2003 \vol 3 \issue 2 \pages 273--333 \mathnet{http://mi.mathnet.ru/mmj89} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2025263} \zmath{https://zbmath.org/?q=an:1046.32004} \elib{http://elibrary.ru/item.asp?id=8379104} 

• http://mi.mathnet.ru/eng/mmj89
• http://mi.mathnet.ru/eng/mmj/v3/i2/p273

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Pratoussevitch A., “Fundamental domains in Lorentzian geometry”, Geometriae Dedicata, 126:1 (2007), 155–175
2. Pratoussevitch A., “On the link space of a Q-Gorenstein quasi-homogeneous surface singularity”, Real and Complex Singularities, Trends in Mathematics, 2007, 311–325
3. Dolgachev I.V., “McKay's Correspondence for Cocompact Discrete Subgroups of SU(1,1)”, Groups and Symmetries: From Neolithic Scots to John McKay, CRM Proceedings & Lecture Notes, 47, 2009, 111–133
4. Pratoussevitch A., “The combinatorial geometry of Q-Gorenstein quasi-homogeneous surface singularities”, Differential Geom Appl, 29:4 (2011), 507–515
5. He Ya.-H., Read J., “Hecke Groups, Dessins D'Enfants, and the Archimedean Solids”, Front. Physics, 3 (2015), 91