Abstract
In this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ < PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.
References
[1] William Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981), no. 8, 574-592.Search in Google Scholar
[2] Alan F. Beardon, A Premier on Riemann Surfaces, London Mathematical Society Lecture Note Series, vol. 78, Cambridge University Press, Cambridge, 1984.Search in Google Scholar
[3] Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983.10.1007/978-1-4612-1146-4Search in Google Scholar
[4] J. Cantwell and L. Conlon, Leaves with isolated ends in foliated 3-manifolds, Topology 16 (1977), no. 4, 311-322.Search in Google Scholar
[5] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass.-London-Sydney, 1978. Reprinting of the 1966 original; Allyn and Bacon Series in Advanced Mathematics.Search in Google Scholar
[6] Henri de Saint-Gervais, Uniformization of Riemann Surfaces Revisiting a hundred-year-old theorem, European Mathematical Society, 2010.Search in Google Scholar
[7] H. M. Farkas and I Kra, Riemann Surfaces, 2nd ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992.10.1007/978-1-4612-2034-3Search in Google Scholar
[8] L. R. Ford, The fundamental region for a Fuchsian group, Bull. Amer. Math. Soc. 31 (1925), no. 9-10, 531–539.Search in Google Scholar
[9] Hans Freudenthal, Über die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931), no. 1, 692–713 (German).10.1007/BF01174375Search in Google Scholar
[10] Étienne Ghys, Topologie des feuilles génériques, Ann. of Math. (2) 141 (1995), no. 2, 387–422 (French).10.2307/2118526Search in Google Scholar
[11] Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.Search in Google Scholar
[12] Svetlana Katok, Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics, Homogeneous flows, moduli spaces and arithmetic, Clay Math. Proc., vol. 10, Amer. Math. Soc., Providence, RI, 2010, pp. 243–320.Search in Google Scholar
[13] Béla. Kerékjártó, Vorlesungen über Topologie I, Mathematics: Theory & Applications, Springer, Berlín, 1923.10.1007/978-3-642-50825-7Search in Google Scholar
[14] John M. Lee, Introduction to topological manifolds, Graduate Texts in Mathematics, vol. 202, Springer-Verlag, New York, 2000.Search in Google Scholar
[15] Bernard Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988.10.1007/978-3-642-61590-0Search in Google Scholar
[16] Dúwang Prada, A golden Cantor Set, Undergrade Dissertation, Industrial University of Santander, Bucaramanga, Colombia, 2006 (Spanish).Search in Google Scholar
[17] C. Ramírez Maluendas and F. Valdez, Veech group of infinite genus surfaces, Algebr. Geom. Topol. 17 (2017), no. 1, 529-560.Search in Google Scholar
[18] Frank Raymond, The end point compactification of manifolds, Pacific J. Math. 10 (1960), 947–963.10.2140/pjm.1960.10.947Search in Google Scholar
[19] Ian Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106 (1963), 259–269.10.1090/S0002-9947-1963-0143186-0Search in Google Scholar
[20] Richard Evan Schwartz, Mostly surfaces, Student Mathematical Library, vol. 60, American Mathematical Society, Providence, RI, 2011.10.1090/stml/060Search in Google Scholar
[21] Ernst Specker, Die erste Cohomologiegruppe von Überlagerungen und Homotopie-Eigenschaften dreidimensionaler Mannigfaltigkeiten, Comment. Math. Helv. 23 (1949), 303–333 (German).10.1007/BF02565606Search in Google Scholar
[22] Stephen Willard, General topology, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970.Search in Google Scholar
[23] Anna M. Zielicz, Geometry and dynamics of infinitely generated Kleinian groups-Geometrics Schottky groups, PhD Dissertation, Universität Bremen, 2015.Search in Google Scholar
© 2020 John A. Arredondo et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
