Fundamental domains for genus-zero and genus-one congruence subgroups

Abstract

AbstractIn this paper, we compute Ford fundamental domains for all genus-zero and genus-one congruence subgroups. This is a continuation of previous work, which found all such groups, including ones that are not subgroups ofPSL(2,ℤ). To compute these fundamental domains, an algorithm is given that takes the following as its input: a positive square-free integerf, which determines a maximal discrete subgroup Γ0(f)+ofSL(2,ℝ); a decision procedure to determine whether a given element of Γ0(f)+is in a subgroupG; and the index ofGin Γ0(f)+. The output consists of: a fundamental domain forG, a finite set of bounding isometric circles; the cycles of the vertices of this fundamental domain; and a set of generators ofG. The algorithm avoids the use of floating-point approximations. It applies, in principle, to any group commensurable with the modular group. Included as appendices are: MAGMA source code implementing the algorithm; data files, computed in a previous paper, which are used as input to compute the fundamental domains; the data computed by the algorithm for each of the congruence subgroups of genus zero and genus one; and an example, which computes the fundamental domain of a non-congruence subgroup.