Abstract
This short article re-examines the interaction between group actions in hyperbolic geometry and low-dimensional topology, focussing in particular on some contributions of Murray Macbeath to the study of Riemann surface automorphisms. A brief account is included of a potential extension to hyperbolic 3-manifolds.
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References
L.V. Ahlfors, Collected Papers (2 vols.) Birkhauser, Boston. 1982.
A.F. Beardon, Geometry of Discrete Groups, Graduate Texts in Math. vol 91, Springer Verlag, 1983.
N. Bergeron, La conjecture des sous-groupes de surfaces [d’apres J. Kahn & V. Markovic]. Séminaire Bourbaki, 64éme année, Juin 2012, no. 1055.
L. Bers, Papers on Complex Analysis (2vols), Editors, I. Kra & B. Maskit. AMS, Providence, R.I. 1998.
R. Fricke & F. Klein, Vorlesungenüber die Theorie der automorphen Functionen. B. Teubner, Leipzig. 2 vols. (1898, 1912).
W.J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface Quart. J. Math. (Oxford), 17 (1966), 86–97.
J. Kahn & V. Markovic, Immersing almost-geometric surfaces in a closed hyperbolic 3-manifold, Ann. of Math. (2) 175 (2012), 1127–1190.
S. Kojima, Isometry transformations of hyperbolic 3-manifolds, Topology & Appls. 37 (1988), 297–307.
Silvio Levy (editor), The Eightfold Way. MSRI Publications 35, Cambridge Univ. Press, 1999.
A.M. Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Assoc. 5(1961), 90–96.
A.M. Macbeath, Discontinuous Groups and Birational Transformations, ‘The Dundee Notes’, in Proceedings of Summer School at Queen’s College, Dundee, July, 1961. Reissued with corrections, Birmingham University, 1979.
A.M. Macbeath, Generic Dirichlet polygons and the modular group, Glasgow Math. J. 27 (1985), 129–141.
A.M. Macbeath, Hurwitz groups and surfaces, in Levy The Eightfold Way, 103–113.
A. D. Mednykh, Automorphisms of hyperbolic manifolds, A.M.S. Transl. (2) vol. 151 (1992), 107–119.
J. Milnor, Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. (New Series), 6 (1982), 9–24.
J-P. Otal, The Hyperbolisation Theorem for fibered 3-manifolds. SMF-AMS Texts & Monographs 7, AMS (Providence RI), 2001.
C.L. Siegel, Some remarks on discontinuous groups, Ann. of Math. (2) 46 (1945), 708–718.
W.P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. A.M.S. (N. S.) 6 (1982), 357–381.
Hermann Weyl, Die Idee der Riemannschen Fläche. B.Teubner, 1913.
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Harvey, W.J. (2016). Discrete Groups and Surface Automorphisms: A Theorem of A.M. Macbeath. In: Širáň, J., Jajcay, R. (eds) Symmetries in Graphs, Maps, and Polytopes. SIGMAP 2014. Springer Proceedings in Mathematics & Statistics, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-30451-9_9
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DOI: https://doi.org/10.1007/978-3-319-30451-9_9
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