Title:
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On the Infinite Loch Ness monster (English) |
Author:
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Arredondo, John A. |
Author:
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Ramírez Maluendas, Camilo |
Language:
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English |
Journal:
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Commentationes Mathematicae Universitatis Carolinae |
ISSN:
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0010-2628 (print) |
ISSN:
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1213-7243 (online) |
Volume:
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58 |
Issue:
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4 |
Year:
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2017 |
Pages:
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465-479 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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In this paper we introduce the topological surface called {Infinite Loch Ness monster}, discussing how this name has evolved and how it has been historically understood. We give two constructions of this surface, one of them having translation structure and the other hyperbolic structure. (English) |
Keyword:
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Infinite Loch Ness monster |
Keyword:
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tame Infinite Loch Ness monster |
Keyword:
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hyperbolic Infinite Loch Ness monster |
MSC:
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51M15 |
idZBL:
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Zbl 06837080 |
idMR:
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MR3737119 |
DOI:
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10.14712/1213-7243.2015.227 |
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Date available:
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2017-12-12T06:48:59Z |
Last updated:
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2020-01-05 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/146991 |
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Reference:
|
[Abi81] Abikoff W.: The uniformization theorem.Amer. Math. Monthly 88 (1981), no. 8, 574–592. Zbl 0482.30034, MR 0628026 |
Reference:
|
[ARM] Arredondo J.A., Ramírez Maluendas C.: On infinitely generated Fuchsian groups of some infinite genus surfaces.preliminary manuscript. |
Reference:
|
[ARMV17] Arredondo J.A., Ramírez Maluendas C., Valdez F.: On the topology of infinite regular and chiral maps.Discrete Math. 340 (2017), no. 6, 1180–1186. Zbl 1369.05101, MR 3624603, 10.1016/j.disc.2016.12.023 |
Reference:
|
[Bea84] Beardon A.F.: A Premier on Riemann Surfaces.London Mathematical Society Lecture Note Series, 78, Cambridge University Press, Cambridge, 1984. MR 1238511 |
Reference:
|
[CC78] Cantwell J., Conlon L.: Leaf prescriptions for closed $3$-manifolds.Trans. Amer. Math. Soc. 236 (1978), 239–261. Zbl 0398.57009, MR 0645738 |
Reference:
|
[CC77] Cantwell J., Conlon L.: Leaves with isolated ends in foliated $3$-manifolds.Topology 16 (1977), no. 4, 311–322. Zbl 0385.57007, MR 0645739, 10.1016/0040-9383(77)90038-6 |
Reference:
|
[CBG08] Conway J.H., Burgiel H., Goodman-Strauss C.: The Symmetries of Things.A K Peters, Ltd., Wellesley, Massachusetts, 2008. Zbl 1173.00001, MR 2410150 |
Reference:
|
[Cox36] Coxeter H.S.M.: Regular skew polyhedra in three and four dimension, and their topological analogues.Proc. London Math. Soc. (2) 43 (1937), no. 1, 33–62. MR 1575418 |
Reference:
|
[FK36] Fox R.H., Kershner R.B.: Concerning the transitive properties of geodesics on a rational polyhedron.Duke Math. J. 2 (1936), no. 1, 147–150. Zbl 0014.03401, MR 1545913, 10.1215/S0012-7094-36-00213-2 |
Reference:
|
[Ghy95] Ghys É.: Topologie des feuilles génériques.Ann. of Math. (2) 141 (1995), no. 2, 387–422. Zbl 0843.57026, MR 1324140, 10.2307/2118526 |
Reference:
|
[Hil00] Hilbert D.: Mathematical problems.Bull. Amer. Math. Soc. (N.S.) 37 (2000), no. 4, 407–436; reprinted from Bull. Amer. Math. Soc. 8 (1902), 437–479. Zbl 0979.01028, MR 1779412, 10.1090/S0273-0979-00-00881-8 |
Reference:
|
[HS06] Hubert P., Schmidt T.A.: An introduction to Veech surfaces.Handbook of dynamical systems, 1B, Elsevier B.V., Amsterdam, 2006, pp. 501–526. Zbl 1130.37367, MR 2186246 |
Reference:
|
[Kat92a] Katok S.: Fuchsian Groups.Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. Zbl 1243.37029, MR 1177168 |
Reference:
|
[KZ75] Katok A.B., Zemljakov A.N.: Topological transitivity of billiards in polygons.Mat. Zametki 18 (1975), no. 2, 291–300 (Russian). MR 0399423 |
Reference:
|
[KMS86] Kerckhoff S., Masur H., Smillie J.: Ergodicity of billiard flows and quadratic differentials.Ann. of Math. 124 (1986), no. 2, 293–311. Zbl 0637.58010, MR 0855297, 10.2307/1971280 |
Reference:
|
[Ker23] Kerékjártó B.: Vorlesungen über Topologie I.Mathematics: Theory & Applications, Springer, Berlin, 1923. |
Reference:
|
[KZ03] Kontsevich M., Zorich A.: Connected components of the moduli space of abelian differentials with prescribed singularities.Invent. Math. 153 (2003), no. 3, 631–678. MR 2000471, 10.1007/s00222-003-0303-x |
Reference:
|
[Lox81] Loxton J.H.: Captain Cook and the Loch Ness Monster.James Cook Mathematical Notes 27 (1981), no. 3, 3060–3064. |
Reference:
|
[Lox83] Loxton J.H.: The graphs of exponential sums.Mathematika 30 (1983), no. 2, 153–163. Zbl 0517.10040, MR 0737174, 10.1112/S0025579300010500 |
Reference:
|
[Ma88] Maskit B.: Kleinian Groups.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 287, Springer, Berlin, 1988. Zbl 0940.30022, MR 0959135 |
Reference:
|
[MT02] Masur H., Tabachnikov S.: Rational billiards and flat structures.Handbook of dynamical systems, 1A, North-Holland, Amsterdam, 2002, pp. 1015–1089. Zbl 1057.37034, MR 1928530 |
Reference:
|
[Mol06] Möller M.: Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve.Invent. Math. 165 (2006), 633–649. Zbl 1111.14019, MR 2242629, 10.1007/s00222-006-0510-3 |
Reference:
|
[MR] Muciño-Raymundo J.: Superficies de Riemann y Uniformización.http://www.matmor.unam.mx/~muciray/articulos/Superficies_de_Riemann.pdf. |
Reference:
|
[Nis75] Nishimori T.: Isolated ends of open leaves of codimension-one foliations.Quart. J. Math. 26 (1975), no. 1, 159–167. Zbl 0303.57012, MR 0377927, 10.1093/qmath/26.1.159 |
Reference:
|
[PS81] Phillips A., Sullivan D.: Geometry of leaves.Topology 20 (1981), no. 2, 209–218. Zbl 0454.57016, MR 0605659, 10.1016/0040-9383(81)90039-2 |
Reference:
|
[PSV11] Przytycki P., Schmithüsen G., Valdez F.: Veech groups of Loch Ness monsters.Ann. Inst. Fourier (Grenoble) 61 (2011), no. 2, 673–687. Zbl 1266.32016, MR 2895069, 10.5802/aif.2625 |
Reference:
|
[RMV17] Ramírez Maluendas C., Valdez F.: Veech group of infinite-genus surfaces.Algebr. Geom. Topol. 17 (2017), no. 1, 529–560. MR 3604384, 10.2140/agt.2017.17.529 |
Reference:
|
[Ric63] Richards I.: On the classification of noncompact surfaces.Trans. Amer. Math. Soc. 106 (1963), 259–269. Zbl 0156.22203, MR 0143186, 10.1090/S0002-9947-1963-0143186-0 |
Reference:
|
[Son75] Sondow J.: When is a manifold a leaf of some foliation?.Bull. Amer. Math. Soc. 81 (1975), no. 3, 622–624. Zbl 0309.57012, MR 0365591, 10.1090/S0002-9904-1975-13764-5 |
Reference:
|
[Spe49] Specker E.: Die erste Cohomologiegruppe von Überlagerungen und Homotopie-Eigenschaften dreidimensionaler Mannigfaltigkeiten.Comment. Math. Helv. 23 (1949), 303–333. Zbl 0036.12803, MR 0033520, 10.1007/BF02565606 |
Reference:
|
[Spi79] Spivak M.: A comprehensive introduction to differential geometry, Vol. I.second edition, Publish or Perish, Inc., Wilmington, Del., 1979. Zbl 0439.53001, MR 0532834 |
Reference:
|
[Ste97] Steuart C.: The Loch Ness Monster: The Evidence.Prometheus Books, USA, 1997. |
Reference:
|
[Val09] Valdez F.: Infinite genus surfaces and irrational polygonal billiards.Geom. Dedicata 143 (2009), 143–154. Zbl 1190.37040, MR 2576299, 10.1007/s10711-009-9378-x |
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