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 Tr. Mat. Inst. Steklova, 2009, Volume 266, Pages 237–262 (Mi tm1874)

Riemann Surfaces with Orbifold Points

L. O. Chekhovabc

a Alikhanov Institute for Theoretical and Experimental Physics, Moscow, Russia
b Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
c Laboratoire J.-V. Poncelet, Moscow, Russia

Abstract: We interpret the previously developed Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces) as the Teichmüller theory of Riemann surfaces with orbifold points of order 2. In the Poincaré uniformization pattern, we describe necessary and sufficient conditions for the group generated by the Fuchsian group of the surface with added inversions to be of the almost hyperbolic Fuchsian type. All the techniques elaborated for the bordered surfaces (quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebra of geodesic functions) are equally applicable to the surfaces with orbifold points.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2009, 266, 228–250

Bibliographic databases:

Document Type: Article
UDC: 515.165.7+517.545

Citation: L. O. Chekhov, “Riemann Surfaces with Orbifold Points”, Geometry, topology, and mathematical physics. II, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday, Tr. Mat. Inst. Steklova, 266, MAIK Nauka/Interperiodica, Moscow, 2009, 237–262; Proc. Steklov Inst. Math., 266 (2009), 228–250

Citation in format AMSBIB
\Bibitem{Che09} \by L.~O.~Chekhov \paper Riemann Surfaces with Orbifold Points \inbook Geometry, topology, and mathematical physics.~II \bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 70th birthday \serial Tr. Mat. Inst. Steklova \yr 2009 \vol 266 \pages 237--262 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm1874} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2603271} \zmath{https://zbmath.org/?q=an:1183.30047} \transl \jour Proc. Steklov Inst. Math. \yr 2009 \vol 266 \pages 228--250 \crossref{https://doi.org/10.1134/S0081543809030146} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000270722100014} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-70350417096} 

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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. M. Mazzocco, L. O. Chekhov, “Orbifold Riemann surfaces: Teichmüller spaces and algebras of geodesic functions”, Russian Math. Surveys, 64:6 (2009), 1079–1130
2. Chekhov L.O., “Orbifold Riemann surfaces and geodesic algebras”, J. Phys. A, 42:30 (2009), 304007, 32 pp.
3. Chekhov L., Mazzocco M., “Shear coordinate description of the quantized versal unfolding of a $D_4$ singularity”, J. Phys. A, 43:44 (2010), 442002, 13 pp.
4. Chekhov L., Mazzocco M., “Isomonodromic deformations and twisted Yangians arising in Teichmüller theory”, Adv. Math., 226:6 (2011), 4731–4775
5. Chekhov L. Mazzocco M., “Teichmüller Spaces as Degenerated Symplectic Leaves in Dubrovin-Ugaglia Poisson Manifolds”, Physica D, 241:23-24 (2012), 2109–2121
6. Chekhov L. Shapiro M., “Teichmüller Spaces of Riemann Surfaces with Orbifold Points of Arbitrary Order and Cluster Variables”, Int. Math. Res. Notices, 2014, no. 10, 2746–2772
7. Chekhov L. Mazzocco M., “Colliding Holes in Riemann Surfaces and Quantum Cluster Algebras”, Nonlinearity, 31:1 (2018), 54–107
8. Labardini-Fragoso D., Velasco D., “On a Family of Caldero-Chapoton Algebras That Have the Laurent Phenomenon”, J. Algebra, 520 (2019), 90–135
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