Convex standard fundamental domain for subgroups of Hecke groups

cited authors

  • Ibrahimou, B; Yayenie, O

fiu authors

abstract

  • It is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2,ℝ), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet's and Ford's polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If Γ2 is a subgroup of Γ1 such that Γ1 = Γ2 · {L1,L2,...,Lm} and double-struck F sign is the closure of a fundamental domain of the bigger group Γ1, then the set R=(∪k=1m L k (double-struck F sign))o is a fundamental domain of Γ2. One can ask at this juncture, is it possible to choose the right coset suitably so that the set R is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups. © Copyright Australian Mathematical Publishing Association Inc. 2010.

publication date

  • February 1, 2011

Digital Object Identifier (DOI)

start page

  • 96

end page

  • 107

volume

  • 83

issue

  • 1