Nonexistence of h-convex cuspidal standard fundamental domain
Bull. Korean Math. Soc. 2009 Vol. 46, No. 5, 823-833
Printed September 1, 2009
Omer Yayenie
Murray State University
Abstract : It is well-known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of the modular group, then its translates by the group elements form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such hyperbolically convex polygons can be obtained by using Dirichlet's and Ford's polygon constructions. Another method of obtaining a fundamental domain for subgroups of the modular group is through the use of a right coset decomposition and we call such domains standard fundamental domains. In this paper we give subgroups of the modular group which do not have hyperbolically convex standard fundamental domain containing only inequivalent cusps.
Keywords : modular group, congruence subgroup, fundamental domain, hyperbolic convexity
MSC numbers : Primary 11F06, 11F03, 20H05, 20H10
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