Convex Standard Fundamental Domain For subgroups of Hecke Groups

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 2 is a subgroup of 1 such that 1 = 2 · {L 1 , L 2 , . . . , L m } and F is the closure of a fundamental domain of the bigger group 1 , then the setis a fundamental domain of 2 . 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.2000 Mathematics subject classification: primary 11F06; secondary 11F03, 20H05, 20H10.